Graph Packing Problems
Sarah Pantel
B.Sc. (Major), University of Manitoba, 1994
A THESIS S U B M I T T E D IN PARTIAL FULFILL-VENT
O F T H E R E Q U I R E M E N T S FOR THE DEGREE O F
MASTER OF SCIENCE
in the Department
of
Mathematics and Statistics
@ Sarah Pantel 1999
SIMON FRASER UNIVERSITY
October 1999
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Abstract
T h e problem of covering the vertices of a given undirected graph with a maximum number
of disjoint copies of the complete g raph o n two vertices, K2, is called the maximum matching
problem. Due t o t h e fruitful results of matching theory, such as the polynomial algorithms
for finding the maximum matching of a given graph, i t was hypothesized t h a t similar results
might hold for the analogous problem using a graph o the r than K2. T h e problem of covering
a graph with copies of graphs o the r than only K2 is called the graph packing problem. Let
G be a family of graphs. Formally, a G-packing of a graph H is a se t of vertex disjoint
subgraphs {G1 , . . . , Cik) of H such t h a t each G; is isomorphic t o some graph in <3. T h e
graph packing problem has been studied for a variety of families G . Although t h e positive
results are not as abundant as for t h e matching problem, many polynomial results have
been found for graph packing problems o n undirected graphs.
The undirected graph packing problem has now been studied quite extensively. However.
this is not t h e case for the directed packing problem. In this thesis, we s t u d y the directed
graph packing problem for families of directed paths , directed cycles and directed stars. T h e
main new result of the thesis is a Berge-like characterization of maximum packings for the
directed graph packing problem with t h e family 6).
Acknowledgments
Firstly, I would like t o acknowledge my senior supervisor, Dr. Pavoi Hell, for his direction
and advice on this thesis. I would also like t o thank my other supervisors Dr. Brian AIspach
and Dr. Rick Brewster for serving on the committee. I am very grateful t o Dr. Brewster
for his invaluable help with the editing of this thesis, with the proof of the main result of
Chapter 3 as well as for his financial support. I would also like t o thank N.S.E.R.C. for the
financial support of a PGS A award throughout most of this degree and the Department of
Mathematics and Statistics for various TA positions.
I would like t o thank my parents for their encouragement and prayers and my parents-
in-law for their support. Lastty, but most importantly, I'd like t o thank my husband.
Christian, for constant encouragement, understanding and hours of proof-reading through-
out the preparation of this thesis and the Lord Jesus Christ for helping me t o persevere
with my studies and for giving me strength and the right perspective.
"Give me, 0 Lord, a steadfast heart, which no unworthy affection may drag
downwards: give me an unconquered heart, which no tribulation can wear out;
give me an upright heart, which no unworthy purpose may tempt aside. Bestow
on me also, 0 Lord my God, understanding to know you, diligence t o seek you,
wisdom t o find you, and a faithfulness tha t may finally embrace you, through
Jesus Christ our Lord."
Thomas Aquinas
"I said, 'You a re my servant'; I have chosen you and have not rejected you. So
d o not fear, for I a m with you. Do not be dismayed, for I a m your God. I will
strengthen you and help you; t will uphold you with my righteous right hand."
Isaiah 4 1:9-10 (NIV)
Dedication
To my husband Christian
Contents
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract 111
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-lcknowledgments iv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication v
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables viii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures is
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1
. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Definitions and Notation 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Problem 5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Complexity Theory 6
. . . . . . . . . . . . . . . . . Survey of the General Graph Packing Problem 9
. . . . . . . . . . . . . . . . . . . . . . . 2.1 Families of Complete Graphs 9
. . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 {K2, Fi)-Packings 18
. . . . . . . . . . . . . . . . . . 2.2 Families of Complete Bipartite Graphs 'LO
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Families of Paths 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Families of Cycles 28
. . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Families of Mixed Graphs 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Applications 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directed Graph Packings 31
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions and Notation 31
. . . . . . . . . . . . . . . . . . . . . . . 3.2 Packing with Directed Paths 32
. . . . . . . . . . . . . . . . . . . . . . 3.2.1 One-Element Families 3'2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main Theorem 42
. . . . . . . . . . . . . . . . . . . . . . . Packing with Directed Cycles 59
. . . . . . . . . . . . . . . . . . . . . . . . 3.5 Packing with Directed Stars 62
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -I Conclusion 68 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography 13
vii
List of Tables
. . . . . . . . . . . . 4.1 Complexity of the packing problem for families of cliques 69
4.2 Complexity of the packing problem for families of complete bipartite graphs . TO
. . . . . . . . . . . . 4.3 Complexity of the packing problem for families of paths 7 0
. . . . . . . . . . . . 4.1 Complexity of the packing problem for families of cycles TO
4.5 Complexity of the packing problem for mixed families . . . . . . . . . . . . . 71
. . . . . . . . . . . . 4.6 Complexity of the packing problem in the directed case 71
. . . . . . . . . . . . . . . . . . . . 3 5 Gadget for the COL-{fi}-factor problem 37
. . . . . . . . . . . 3.6 A COL.{~?, }.factor of the gadget including the connectors 37
. . . . . . . . . . . 3.7 A COL-{fi}-factor of the gadget excluding the connectors 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Placement of f i 39
. . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Gadget for path factor problems 40
. . . . . . . . . . . . . . . . . . . . . . . 3.10 X G-Factor including the connectors 41
. . . . . . . . . . . . . . . . . . . . . . . 3-11 A G-Factor excluding the connectors 41
. . . 3.12 Examples of augmenting configurations for the {fi, problem 4.5
. . . 3.13 Augmenting configuration components for the (6. f i ~ - ~ a c l i i n ~ problem 46
. . . 3.14 Examples of augmented configurations for the { f i , fi}-packing problem 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Example diagrams for case 2b 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Example diagrams for case 2c 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Example diagrams for case 3a 57
. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Example diagrams for case 3b 58
. . . . . . . . . . . . . . . . . . . . . . . 3.19 Gadget for t h e {c3)-factor problem 60
. . . . . . . . . . . . . . . . . 3.20 A {c;}-factor containing the connector vertices 60
. . . . . . . . . . . . . . . . . 3.21 A {c3}-factor excluding the connector vertices 61
. . . . . . . . . . . . . . . . . . . . . . . 3.22 Gadget for the {c;}-factor problem 62
. . . . . . . . . . . . . . . . . 3.23 A {c4)-factor containing the connector vertices 63
. . . . . . . . . . . . . . . . . 3.24 A {c4}-factor excluding the connector vertices 64 - . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Gadget for the {S2)-factor problem 64 - . . . . . . . . . . . . . . . . . . . . . . . . 3.26 Gadget for the {S3}-factor problem 65
. . . . . . . . . . . . . . . . 3.27 An {&)-factor containing the connector vertices 66 - . . . . . . . . . . . . . . . . . 3.28 .4 n {S3)-factor excluding the connector vertices 66 - . . . . . . . . . . . . . . . . . . . . . . . . 3.29 Gadget for the (5';)-factor problem 67
Chapter 1
Introduction
T h e aim of this thesis is two-fold. We will survey results on the general graph packing
problem for undirected graphs and then we will begin to develop a similar theory for directed
graphs.
1.1 Definitions and Notation
In order to understand the terminology of t h e g raph packing problem, we must first introduce
some definitions from graph theory and from matching theory.
A simple graph, or just gruph, H is a n ordered pair ( V ( H ) , E ( H ) ) , where V ( H ) =
{vl , . . . , v,) and E ( H ) = ( e l , . . . , e m ) . T h e elements v ; , i = 1 . . . n, of V ( N ) are called
vertices and t he elements e;: i = 1 . . . m , of E ( H ) are called edges. Each edge e; is an
unordered pair of vertices. These two vertices a r e incident with the edge and adjacent to
each other. T h e degree of a vertex is t h e number of edges with which i t is incident. T h e
size of t he vertex-set of a graph, IV(H)I , is called the order of the graph.
A visual representation of this graph is given in Figure 1.1.
.4 directed graph or digraph, D, is defined by t h e ordered pair ( V ( D ) , A ( D ) ) . Here,
CHAPTER 1. INTRODCiCTIOrV
Figure 1.1: H = ( V ( H ) , E ( H ) )
I f ( D ) , is the set of vertices of D, a n d A ( D ) is t h e se t of arcs o r directed edges of D. Each
arc is a n ordered pair of vertices. T h u s if e; = vjvk is a n a rc , then t h e directed edge e;
begins in vertex v, and ends in vertex vk.
A complete gmph, o r clique, on n vertices, I<,? is a graph with vertex-set { u l , . . .: u,}
and edge-set {vivj li # j ) . A complete bipartite graph, K,,,, is a graph with vertex-set
{ u l l . . ., V , } U { ~ ~ , . . ., un} a n d edge-set {viujli E (1,. . .. rn). j E (1 , . . ., n}}. The complete
bipart i te graph KL,, is called a star o n n edges a n d is denoted S,. A path of length k.
denoted Pk, is a graph with vertex-set (uo, v ~ , . . . , uk} and edge-set ( e l , . . . . el;) such t h a t
e; = V;_~L';. A (u, v)-path is a pa th such t h a t t.0 = u and uk = u. Finally, a cycle of length
I ; , denoted Ck, is a graph with vertex-set {Q, . . ., v k ) a n d edge-set { e l , . . . , e k ) such t h a t
e; = v;v;+l, i = 1,. . . k - 1, and ek = vkul. An example of each of these graphs is shown in
Figure 1.2.
Figure 1.2: Examples of commonly used gra.phs
W e will now introduce s o m e necessary definitions from matching theory. Consider a
graph H = (V, E ) . A graph I< = (V', E') such t h a t V' C V and E' 5 E is called a subgmph
of H a n d we write It* C - W. A matching M of a graph H is a s e t of pairwise vertex disjoint
C'HA PTER 1. INTRODUCTION
edges of H . -4 vertex of H which is incident with an edge of hl is cocered by ibl and is
said t o be in the matching M . T h e o t h e r vertices a re called uncovered by Ad, o r ezposed in
:1/. A path in H is a subgraph A- of H such t h a t Ii is a path. .An allernatiny pdh in H .
with respect t o a matching 1C.l in H. is a pa th in H whose edges are alternately in and o u t
of the matching M . A n augmenting path is a n al ternating path which begins and ends in
esposed vertices. T h e size of a matching is t h e number of vertices covered by the matching.
A matching of largest size is called a mazimum matching. If a matching covers all vertices
of N, then i t is a perfect matching o r I-/actor of H . A perfect matching of Cs is shown in
Figure 1.3 where the bold edges represent edges in the matching.
Figure 1.3: A perfect matching of Cs
In matching theory, we usually search for maximum matchings o r 1-factors of graphs.
A result established by Berge [8] says t h a t a matching M of a graph H is maximum if a n d
only if H contains no augmenting p a t h with respect t o M . Edmonds uses the search for
augmenting paths approach t o find a n algorithm for maximum matchings in general graphs
[20]. A theorem of Hall 1261 for bipart i te graphs gives a necessary and sufficient condition
for the existence of a matching covering each vertex in one of the parts. Let H be a bipartite
g raph with bipartition (X, Y) and let N ( S ) denote t h e neighbourhood of t h e se t S C X. This neighbourhood is t h e set of all vertices of H which a re adjacent t o vertices of S. Hall's
theorem s ta tes t h a t t h e bipartite g raph H contains a matching which covers all vertices of
-Y if a n d only if IN(S)I > IS1 for all S C X. A necessary and sufficient condition for t h e existence of a perfect matching of a n arbitrary
g raph was found by T u t t e [45]. Before we give th is condition, we introduce some necessary
definitions. A non-empty graph H is connected if for all u , u E V there exists a (u, v)-path.
An induced subgmph S of H is a subgraph of H with vertex set V ( S ) C V ( H ) and edge set
E (S) 5 E ( H ) such t h a t E ( S ) is made up of those edges of E ( H ) incident only with vertices
in t h e vertex set V ( S ) . A subgraph S of H is a component of H if S is connected a n d there
C'HA PTER I. INTRODUCTION
does not exist an edge ut. where u E 17(S) a n d v E V ( H ) \ I'(S). .A component is odd if
t h e number of vertices in t h e component is odd. Now consider t h e g raph H \ S obtained
from H by removing all vertices of t h e se t S C V ( H ) and all edges incident with those
vertices. T u t t e proved t h a t a g r a p h H has a perfect matching if a n d only if the number of
o d d components of H \ S is less than o r equal t o (SI for all S 2 V ( H ) .
T h e richness of these results from matching theory suggests t h a t expanding t h e theory
may also be fruitful. Let G = (GI . . . , Gk) be a family of graphs. .A G-packing of a graph
H is a set of pairwise vertex-disjoint subgraphs of I1 each of which is isomorphic t o some
graph in t h e family G . .A vertex of I1 is covered by the G-packing if i t belongs t o a subgraph
of t h e G-packing. If G = (G), then we call a G-packing a G-packing. Since there is a natura i
correspondence between t h e edges of a graph a n d the se t of subgraphs isomorphic t o K2, i t
is clear t h a t every I<*-packing of a graph H corresponds to a matching of H and vice versa-
T h e size of a packing is the number of vertices of W covered by t h e packing. A G-packing of
H of largest size is called a mazimum G-packing of H. If the G-packing covers all vertices
of H , then it is a perfect G-pck ing o r a G-factor of H.
1.2 Background
T h e founders of matching theory a r e generally agreed t o be Jul ius Petersen (1839-1910)
and Denes Konig (1884-1944). Petersen is best known for his results o n regular graphs and
Konig for his results on bipart i te graphs. Other mathematicians instrumental in matching
theory were Euler, Kirchhoff a n d Tait. .4 good history of the work of Petersen, Konig and
their contemporaries can b e found in t h e preface of L o v k and Plummer's book Matching
Theory [42]. An history of g raph theory from 1736 t o 1936 c a n b e found in [lo], a n d a
"Sampler of major events of matching theoryn can be found in [43].
Matching theory has many interesting applications in areas such as scheduling problems
[27], assignment problems [Xi, 331, t ranspor ta t ion problems a n d network flow problems
[23, 331, the shortest p a t h problem [33], t h e travelling salesman problem [13, 2.51, a n d t h e
Chinese Postman problem [21]. Results from packing theory have useful applications to
code optimization [ l l ] , clustering [32], a n d component placing [34]. Hell and Kirkpatrick
considered one such application in [36] where they write t h a t their interest in the s t u d y of
generalized matchings s t emmed from t h e s t u d y of scheduling examination periods.
CH.4 PTER I . IXTROD UCTION
1.3 The Problem
In th is thesis, we s tudy 6-packing problems and G-factor problems. We have s ta ted t h a t
the theory of packings is a n extension of matching theory. T h e maximum matching problem
is t h e problem of searching for a matching M of a given input g raph H . such t h a t :\I ccvers
a mas imum number of vertices. T h e analogous problem in packing theory, t h e maximum
G-packing problem, which we will refer t o simply as the G-packing problem, is t h e problem
of searching for a maximum G-packing of a n input graph H . Many polynomial algorithms
es is t for solution of t h e maximum matching problem, [20, 19, 2'2,421, a n d i t will b e shown in
C h a p t e r 2 t h a t many polynomial algori thms have also been found for t h e G-packing problem
for various families G. In matching theory, we s tudy a special case of the maximum matching problem called
t h e perfect matching problem. Th i s is t h e problem of finding a p e r f s t matching of the input
g raph H. Formally, a decision problem is a function from t h e instances of a problem t o the
answers "yes" a n d "no". For example, t h e perfect matching problem is formally defined as ._-_ a decision problem below:
Perfect Match ing: Instance: A graph H
Question: Does H admit a perfect matching, t h a t ist is there
a matching of H of size I V ( H ) I?
T h e packing problem analogous to t h e perfect matching problem is the G-factor problem.
I t c a n be s t a t e d formally as a decision problem as follows:
Given t h e family of graphs G?
G-fac tor: Instance: A graph H
Question: Does H admi t a G-factor, t h a t is, is there a packing of H
using graphs from the family 6 such t h a t the size of the
packing is IV(H) I?
In Chap te r 2, we survey results for t h e G-packing problem. In Chap te r 3, we will s tudy
t h e 6-factor problem in the directed case. This s tudy is from t h e point of view of complexity
theory, which is introduced in the following section.
C'H.4 PTER 1 . INTRODUCTION
Complexity Theory
To begin, we will define three broad complexity classes: the class P: the class AeP; the class
co - JVP. Complexity theory is a fundamental theory of computing science and as such a
fd1 introduction is beyond t h e scope of this thesis. This section serves as an introduction t o
the areas of complesity theory needed for this work. For a full introduction t o complesity
theory, the reader is referred to p23].
A problem is said t o be in t h e class P if it can be solved by a polynomial time deter-
ministic algorithm. Polynomial t ime means t h e number of steps executed by t h e algorithm
is bounded by a poiynomial in t h e size of t h e input. For example, a reasonable measure
for the size of a n input graph might be t h e number of vertices o r the number of edges.
Deterministic means t h a t t h e i + lS t s t e p of the algorithm is uniquely determined by the
first i steps.
T h e class AfP contains decision problems which can be solved in polynomial time by
a nondeterministic algorithm. -4 nondeterministic algorithm has two stages: the guessing
s tage and the checking stage. There a re two possible outcomes of a nondeterministic al-
gorithm. Either t h e instance of the decision problem is a yes-instance ( the answer t o the
question is yes), in which case t h e checking s tage of the algorithm s tops and answers yes,
o r else the instance is not a yes-instance, in which case! the algorithm may s t o p and answer
no, o r i t may never stop. For example, consider a nondeterministic algorithm for the perfect
matching problem. Let the input graph H have IV(H)I = n a n d let i = 1,. . . , n/2. In the
guessing stage, choose a n edge of H a t each of the i steps of the algorithm. This gives 4 edges a n d this guess at a solution is called a certificate. T h e certificate must be polynomial
in size. In t h e checking stage, check whether o r not these 2 edges form a perfect matching.
Th is par t of the algorithm must complete in t ime polynomial in the size of t h e input. In our
esarnple, determining whether o u r certificate is a perfect matching of H can be achieved in
polynomial time. A yes-certificate of a problem is a certificate which upon testing gives the
answer yes to the decision problem question. In our example, a yes-certificate is a perfect
matching. An instance of the problem is called a yes-instance if there exists a yes-certificate.
Any decision problem solvable by a polynomial ti me deterministic algorithm is solvable
by a polynomial t ime nondeterministic algorithm since we can simply use the deterministic
algorithm as the checking s tage of the nondeterministic algorithm with a n empty guess.
Therefore, t h e problems in t h e class P a r e also in the class n/P a n d so P S J ~ P .
T h e class co - jVP contains t h e complernenl problems. nC, t o those problems ll in the
class ,VP. Formally, if n denotes a decision problem, then co - ,tip = {llcln E N P ) . T h e
problen~ l l c is the s a m e problem a ll but with the answers reversed. -4s explained in ['L-I].
'-If the problem 'Given I , is X t rue for I?' can be solved by a polynomial t ime deterministic
algorithm, then so can the complementary problem 'Given I , is X false for I?' This is
because a deterministic algorithm halts for all inputs, so all we need t o d o is interchange
the 'yes' and 'no' responses". -4 polynomial transformation from one decision problem .4 to
another decision problem B is a function f from the instances a of A to the instances b of R
such t h a t this function can be computed in polynomial t ime and for all instances of A, t h e
instance is a yes-instance if a n d only if f (a), is a yes-instance of B. I t is no t clear t h a t all
problems can be polynomially transformed t o their complement problems. Many problems
in co - jVP d o not seem t o be in N P and so i t is conjectured t h a t JVP # co - JVP. It
has also been conjectured t h a t P = NP f~ co - n/Pt but this conjecture remains a n open
problem.
In this thesis, we will prove A'P-completeness of various packing problems. In order
t o define what is meant by AfP-complete, we must introduce the concept of reducing one
problem t o another- Consider the following decision problems A and B:
A: Instance: a
Question: Is a this way?
B: Instance: b
Question: Is 6 t h a t way?
We will be using t h e term polynomial time reduction, o r simply reduction in this thesis.
This technique is defined in [24 as follows: "The principal technique used for demonstrating
t h a t two problems a r e related is t h a t of "reducing3 one to the other, by giving a constructive
transformation t h a t maps any instance of the first problem into an equivalent instance of
the second. Such a transformation provides t h e means for converting any algorithm t h a t
solves the second problem into a corresponding algorithm for solving t h e first problem.''
Therefore, if we have a polynomial t ime reduction from problem A to problem B, then a
polynomial-time algorithm for B implies tha t there is a polynomial-time algorithm for A.
We denote a polynomial t ime reduction of problem .4 t o problem B by A a O.
Now we can define t h e complexity class of m - c o m p l e t e problems. A problem is AfP-
complete if it is in t h e class NP a n d if all other problems in the ciass N P can b e polynomially
reduced t o it. T h e foundations of n/P-completeness theory were [aid in [14]. Stephen Cook
emphasized the importance of polynomial time reductions. He gave t h e first proof t h a t the
CHAPTER I . IIVTRODUCTION
satisfiability problem (SAT) had the property t h a t every o ther problem in the class A f P
could be polynomially reduced t o it. This property was found to be t rue of a number of
problems a n d became known as ~ffP-complete. T h e class of ArP-complete problems is a
subset of JVP containing the hardest problems of t h e class JVP. Considering Cook's result.
in order t o prove JVP-completeness, we d o no t have t o reduce every problem in iVP t o o u r
problem. Instead, we use t h e following theorem.
Theorem 1.4.1. [24] Let ill and n2 be problems- If I l l a 112 and 112 E JVP? then I l l is
,VP-complete implies F12 is n/P-complete.
Thus, to prove t h a t a problem 112 is NP-comple te , we first show t h a t it is in JVP
and then show t h a t a known NP-complete problem il 1 can be polynomially reduced t o t h e
problem l12. There a r e various ways t o d o this reduction in order t o prove JVP-completeness.
T h e three techniques which a re most frequently used a r e local replacement, restriction, a n d
component design [24]. In th is thesis, we will use t h e local replacement technique.
Finally, a problem ll is in t h e class M P - h a r d if some NP-comple te problem can be
polynornially reduced to it.
Chapter 2
Survey of the General Graph
Packing Problem
T h e G-packing ~ r o b i e r n has been extensively studied over the last few decades. In this
survey, we will consider results o n the general undirected g raph packing problem which asks
"How many vertices of H can we cover with copies of graphs in the family G?". in addition
to the general graph packing problem, many other packing problems have been defined and
studied.
In this chapter, we will separate results according t o t h e type of family G considered. We
begin by considering families of complete graphs and then move o n to families of complete
bipartite graphs, of paths, of cycles, and finally to mixed families. T h e results in this chapter
can be found in (4, .jl 15, 17, 24, 27, 28, 29, 30, 36, 38, 39, 401.
2.1 Families of Complete Graphs
Consider the graph packing problem for G = {Kt). This is clearly polynomial since for any
graph, the set of vertices is a perfect {Kt)-packing. T h e G-packing problem for G = {K2)
is also polynomial since this is simply t h e maximum matching problem.
T h e next family to consider is G = { I c 3 } . This is sometimes referred t o as the triangle
packing problem and i t is NP-complete [ l i ] . T h e NP-completeness proof can be done
by reduction of t h e %dimensional matching problem (3DM) t o the {Ic3}-factor problem as
shown below. T h e problem 3DM is a n NP-complete decision problem which looks for a
C'HA PTER 2. SURVEP' OF T H E G'ENERAL CZ'RA PH P.-\CIiINC; PROBLEM 10
3-dimensionat matching of a s e t of triples T. This problem can be s t a t ed as follows:
3 D M : Instance: An integer q. s e t s W. -3-, and ).' with 11371 = 1-X-I = IYI = q. a n d
a s u b s e t ?-of M'x X x Y.
Question: Is there a subse t M of T containing q triples n o two of which
use t h e s a m e element from a n y of the se ts W , X- o r Y?
T h e next result is s ta ted in [29] a n d we give o u r own proof below.
Theorem 2.1.1. The {li3)-factor problem is J\~P-complete.
Proof: T h e {K3)-factor problem is in AfP since given a potential {K3)-factor
of a g raph H as a certificate, w e c a n verify in polynomial t ime t h a t all vertices
of H a r e covered by exactly o n e K3. We will reduce the AfP-complete problem
3DM t o t h e (IT3}-factor problem t o establish NP-completeness.
Let 7 = ((ti, t * , ts)ltL E W, t2 E .Y, t3 E Y ) b e some subset of FV x X x Y where
W? X, Y = (1,2, . . . , q}. We shall cons t ruc t a graph G such t h a t t he s e t o f triples
7 has a 3DM if and only if G a d m i t s a (lie3)-Factor.
In NP-comple teness proofs we will often use t h e local replacement technique.
In th is technique, we use gadgets to cons t ruc t a n instance of o u r problem. In
th is proof we will use t h e gadge t shown in Figure 2.1. This gadget is used t o
cons t ruc t a graph G.
Figure 2.1: G a d g e t for the {K3)-factor problem
Construction: Consider a g r a p h having vertex se t W u X u Y. We cons t ruc t
the edge se t ,as follows. For each triple ( t 1. t2, t3) of T, identify the connector
oertices, a . 6 , and c, of a copy of t h e gadget with the vertices t l . t 2 and t 3 ,
respectively. of the graph. All o ther vertices, the interior vertices, of each copy
of t h e gadget become vertices and the edges of t h e copies of the gadget become
edges of t h e graph. CaIll the resulting g raph G.
Claim 2.1.1. The set 7 has a three dinrensional matching v a n d only if G has
a {K3}-factor.
Proof: Suppose T has a 3DM. Then for triples ( t l , t2 , t3) in the match-
ing, t h e gadget with connector vertices a = t l , 6 = t2 and c = t3 has
t h e {Ii-3)-factor shown in Figure 2.2.
Figure 2.2: A {K3)-factor containing t h e connector vertices
For triples not in t h e matching, t h a t i s when t l , t z , and t3 a r e each
covered by other triples, the copy of t h e gadget having vertices t l , t2
and t3 has t h e {[GI-factor of Figure 2.3.
Thus, there is always a (K3)-factor of G when there is a 3DM of 7.
Now, suppose G has a {Ic3)-factor. Consider some copy of the gadget.
The point z3 must either be covered by t h e triangle 21x223 o r else by
t h e triangle y323x3. Suppose zs i s covered by y32323. Then in order to
cover t h e vertex 21, we must choose t h e triangle az lxz . Similarly, we
must choose czlz2 and byly2 and we have a {&)-factor of the gadget
as in Figure 2.2. In this case, since t h e connector vertices are covered
CH.4 PTER 2. SURVE)' OF T H E G E X E R A L GRAPH P.~CI\'ILV(: PROBL EILI
Figure 2.3: A {Ic3)-factor excluding the connector vertices
by triangles within the gadget, we choose the triple (a , 6, c) t o be in
the 3-dimensional matching of T.
Now, suppose that 23 is covered by the triangle ~ ~ ~ 2 x 3 . Then vertex y3
must be covered by the triangle yly2g3 and the vertex z3 by the triangle
~ 1 2 2 ~ 3 . In this case, the connectors are not covered by triangles in t he
gadget and so we do not choose (a, 6, c) as a triple of the 3-dimensional
matching.
The resulting set of chosen triples T = { ( a . by c ) ) is a 3-dimensional
matching. 0
Since 3DM is a n n/P-complete problem and we have shown that it reduces t o
the {Ic3)-factor problem, the (Ii-3)-factor problem is also JVP-complete. CI
Hell and Kirkpatrick [35, 361 have proven tha t the problem of packing a n input graph
with a connected graph other than K t o r K2 is NP-complete. However, when we consider
packing with a graph G which is the disjoint union of o copies of KI and P copies of Am2
then the G-packing problem is again polynomial. Consider t he following theorem:
Theorem 2.1.2. [36] Let G be the disjoint union of a copies of I<-i and P copies of K2- Then H admits a G-factor if and only if ( V ( G ) ( divides ( V ( H ) I and some matching of H has at least ,!3 edges.
P m f : Suppose H admits a G-factor. If (V(G)I does not divide IV(H)I then
we cannot cover all vertices of N with copies of G such tha t any vertex of H is
CH.4PTER 2. SURVEY OF T H E GEiVER4 L GR4 PH PAC'IilNC PROBLEM 1 3
in exactly one copy of G'. Therefore, I V(G)I divides II'(H)I. Within the set of
vertices of H used by each copy of G of t h e packing, there needs to be 0 copies
of K2 since G has this number of K2's. Th i s is true for each of the disjoint copies
of G, of which there are \%I, and so in total, the number of edges of some V ( H ) matching of H must have a t least @{V(C)/ edges. Conversely, suppose IV(G) I
V ( H ) divides ]V(H)I and in addition, some matching M of H has at least flfm/ edges. Then we cover vertices of H in t h e following way: choose edges from
:I# and some a vertices from H \ M and let this be a copy of G in the C-packing. I V ( W These chosen edges and vertices cannot be used again. Since there a re fl1t/OIl
edges in t he matching, we can define {w} copies of G in this way. Clearly, the
G-packing is perfect and so this defines a G'-factor of H. 0
For graphs G satisfying the hypothesis of Theorem 2.1.2, the algorithms used to find a
maximum matching can be used t o determine if H admits a G-factor. Since determining
the hypotheses of Theorem 2.1.2 can be done in polynomial time, we have the following
corollary:
Corollary 2.1.1. [36] If G is the disjoint union o f 0 copies of ITI and /3 copies of Ka, then
the G-factor problem is polynomial.
It is also proven in 1361 t h a t for all other g raphs G, the G-factor problem is NP-complete .
The packing problem with G = {K2, K 3 ) has been studied quite extensively, see [ l i . 291:
and has been found to be polynomial. Hell a n d Kirkpatrick t reat the {K2, K3)-packing
problem in C29] and prove the following theorem.
Theorem 2.1.3. [29] Let G = {Am2, K3}. Then there exists a polynomial-time algorithm to
find a maximum size G-packing o j arbitrary input gmph H .
In the proof of Theorem 2.1.3, the authors use Lemma 2.1.1 below which is analogous
to the result from matching theory t ha t a matching of H is maximum if and only if H has
no augmenting path. Before we state the lemma, some more definitions are required. First ,
consider G a {K2, K3)-packing of a graph H. Edges of H which are in the packing a re
called G-edges and subgraphs of H in the packing which a re isomorphic to K3 are called
C'HAPTER 2. SURVEY OF THE GENERAL GRAPH PACIiiNG' PROBLEM 14
Path * * * -@ vo "1 v 'n-I n v n+ 1
Tail "0 "1 "2 n+3 . .
Kite
Figure 2.4: .4ugmenting configurations for t h e {K2, K3)-packing problem
G-triangles. An ultemaling path of H with respect t o G is a pa th in H whose edges a r e
alternately in G a n d not in G. T h a t is, if t h e first edge of t h e p a t h is in the packing, then
the next edge of t h e pa th is not in the packing, e t cetera. An augmenling path of H with
respect to G is a n al ternating path of H with respect to G between two exposed vertices
of H. An augmenting tail of H with respect to G is a n al ternating path of H with respect
to G between a vertex vo which is exposed in G a n d a vertex vn+l which is covered in G ,
such t h a t un+l is covered by t he G-triangle ~ ~ + l t ' ~ + 2 ~ ~ + 3 . Finally, an augmenting kite of
H with respect to G is a n even length al ternating pa th of H with respect t o G between
an uncovered vertex uo a n d some vertex vn covered by the G-edge v,,lvn, together with
a IC3 VnUn+l vn+k+l, and a n alternating p a t h v,+l, . . . , v,+k+l- A degenemte kite i s a kite
with n = 0 a n d k = 1. T h a t is, a triangle with one edge covered by the packing. These
augmenting configurations a r e shown in Figure 2.4. Note t h a t circled vertices denote vertices
which a r e exposed with respect t o the packing.
Augmentation of these configurations is d o n e in t h e following manner. For a n augment-
ing path , we use a switching technique in which all G-edges a r e removed from G and non
G-edges a r e made G-edges. Th i s switching technique is also used to augment t h e tail along
the pa th t.0,. . . , v,+l. In addition, in t h e case of t h e tail, edges vn+lvn+2 and un+lvn+3
become non-packing edges, b u t edge un+zvn+s remains a packing edge. T h e kite is aug-
mented by switching along t h e paths VO, . . . , vn, a n d v,+l , . . . , Un+k+l, and by adding t h e
C'H.4PTER 2. SURVEY OF THE GENERAL C;RAPH PACIilNC: PROBLEM
1i3 v,vn+l v,+k+l to G.
Sow we present the !emma.
Lemma 2.1.1. r29] A {1<*, I<3)-pcking C of a yrnph H is of rnazimum size i j and only if
it admits no augmenting path, tail or kite.
Proof: Clearly, if G is of maximum size then there can be no augmenting
configurations.
Suppose H is a smaliest graph which admits a packing G which has no augment-
ing configurations and another packing G' of II which covers more vertices than
G. Then there must be a vertex v which is covered by G' but not by G. We will
consider two cases:
1. T h e vertex v is covered by a G'-edge vw . Since G has no augmenting configuration, v cannot be on a n edge with a
vertex which is exposed in G , and so w must be covered by G. Since there
a r e no augmenting configurations, w cannot be covered by a G-triangle and
so must be covered by a G-edge wu.
Consider the graph fi with ~ ( f i ) = V ( H ) \ {v, m ) and E( H) = E ( H ) \ { edges incident with u or m), and the packings C and G' restricted to l?.
In fi, G covers two less vertices, namely m and u, and G' covers two less
vertices, namely v and w. Therefore, G' covers more vertices of K than
G and so by t he choice of H, G must admit an augmenting configuration
in H. If t h i s augmenting configuration begins at a vertex different from u
and does not involve u, then i t would be a n augmenting configuration of
G in H as well, but this contradicts the definition of G. If the augmenting
configuration begins a t a vertex other than u and ends a t u, then it is an
augmenting configuration ending at v in H. If t he augmenting configuration
begins a t u, then in H we would have a n augmenting configuration of G
beginning at v with the edge vw and followed by the G-edge wu. -4gain
this contradicts the definition of G.
2. T h e vertex u is covered by a Gf-triangle vwt. Since G has no augmenting
path o r tail, we cannot have any of the graphs of Figure 2.5.
CHAPTER 2. SURVEY OF T H E GENER-4L C;R=ZPH PACKIiVC PROBLEM
Figure '2.5: Vertex t. covered by a Gt-triangle
Figure 2.6: Vertices w and 2 covered by G'-edges
Therefore, both w and z must be covered by G-edges. However, u: and z
cannot be incident with t he same G-edge since G has no degenerate kites.
Thus, there a re two dist inct G-edges as in Figure 2.6.
Consider the graph fi with ~ ( f ? ) = V(H)\{o, to, z } , and E ( H ) = E ( H ) \ {
edges incident with v , w , or z), and the packings G and G' restricted to
a. In fi, G covers four less vertices than in H and G' covers three less
vertices. Therefore, G' covers more vertices of than G. So, by the choice
of H , G must have a n augmenting configuration in H. If t h e augmenting
configuration is a pa th between z and y, then in H we would have the
augmenting configuration of G shown in Figure 2.7, which is a n augmenting
kite in H. This is a contradiction with the definition of G.
If the augmenting configuration begins at x (or y), then i t is a n augmenting
configuration of G beginning a t u in N, which again contradicts the choice
of G. If the configuration begins at some point o ther than x or y, then i t
is an augmenting configuration of G in H as well, another contradiction.
Therefore, the packing G must be of maximum size. C3
T h e augmenting configurations of Lemma 2.2.1 can be used in a n algorithm to find
C'Hk\PTER 2. SURVE k' OF THE C;ENERA L GR.4PH P..1CIiI!VG PROBLEl\d 17
Figure 2.7: Augmenting pa th between x and y
a masimum size packing in polynomial time. Before we give the algorithm, we neecl the
foIlowing proposition which uses some new terminology. A set of vertex disjoint triangles of
H will be denoted T.
Suppose t h a t a given packing G is not of maximum size. Then i t follows from Lemma 2.1.1
tha t G a d m i t s a n augmenting path, tail o r kite a n d therefore there is a larger packing C'
whose se t of triangles differs by a t most one triangle from the set of triangles of G. This is
written formally in the following theorem.
Proposition 2.1.1. ['29] Let G be a {Kz, K3)-packing o/ H, 7 the set of C-tr iangles . Then
G is a {Kz, Ks)-packing of maximum size if and only if no {K2, Ks)-packing G' of H cocers
more vertices than G where the set of G'-triangles:
I . is equal to T:
2. is equal to T \ { u v ~ w ) for some lnbngle {uvw) E T:
3. is equal to T U { u v w ) for some triangle (uvw) T.
Proof: If G is of m a ~ i r n u m size, then clearly there is no packing G' covering
more vertices t h a n G.
Suppose t h a t G is not of maximum size. Then by Lemma 2.1.1 there must be a n
augmenting configuration resulting in a larger packing G'. If t h e configuration
were a pa th , then G' would have the same number of triangles as G (case I) .
If t h e configuration were a tail, then G' w o d d have one less triangle than G
(case 2). Finally, if the configuration were a kite, then G' would have one more
triangle than G (case 3). 0
Proposition 2.1.1 and Lemma 2.1.1 lead to a polynomial-time algorithm for finding a
maximum {K2, K3)-packing. In this algorithm, G denotes a {K2, &)-packing of H , and 7 denotes the set of G-triangles.
C'HA PTER 2. SURVEY OF THE GENERAL GRAPH P.4CIifiVG PR0BLEA.I
Algorithm 2.1.1. To find a mazimum size { I C 2 , K3)-pcking of a gmph H . begin
G':= maximum matching of H
I-:= 0 while (mazimum size of a {I<*, K3) -packing C' with set Tf o j GI-triangles, where T' =
7 \ { u v w ) for some { u v w ) E 7, or Tr = T U {uu.w} jor some { u v w ) # 7) > size of G' do
G':= G'
T:=T return G
end
In this algorithm, the se t of triangles, T' of t h e packing G' is related to t h e set of triangles
'J of G as follows: 7' = 7\ {uvw), where uvw E 7, o r T = T u { u v w ) , where uvw @ 7.
T h e se t T' a n d consequently the packing G', can be found in one of two ways. Firstly, we
systematically choose one triangle uvw from 'J a n d remove it from T t o ob ta in T-. We
then run a maximum matching algorithm o n H \T-. T h e resulting matching together with
T- gives us a packing. If this packing covers more vertices than G, then it is a packing G'
a n d T- = T'. We then update G a n d 7 a n d continue. If not , we choose a different triangle
of T t o remove a n d proceed as before. If n o larger packing G' is found, we move o n to the
triangles which a r e not in T. This t ime, we systematically a d d a triangle uvw which is not
in 7 t o 7 t o obta in T+. Then as before, we run a maximum matching algori thm o n H \ T+ and t h e resulting matching along with 'T+ gives u s a packing. If this packing is larger than
G , then it is a packing G' and T+ = T'. We then update G a n d T and run t h e while loop
again. If no packing G' is found which i s larger than G , then G is a maximum packing a n d
we a r e done. Th i s process is polynomial, since if n = V ( H ) , t h e number of triangles in T
a t any t ime is at most order n, as n o vertices can be repeated, t h e number of triangles not
in 7 at any t ime is at most order n3, a n d finding a maximum matching of t h e remaining
vertices can be found in polynomial t ime [20].
Proposition 2.1.1, Lemma 2.1.1, and t h e algorithm can be extended t o o the r cases including
the case of Theorem 2.1.4 below where G = {K2, F l y . . . , Fk) and all the Fi a r e hypomatch-
able. A graph H is hypomatchable if H has n o perfect matching, b u t H \ v does have a
CH.4 PTER 2. SURVEY OF THE GENERAL C;RA PN P.4CKIiVG PROBLEM
Figure 2.8: T h e kite for G = {Ir;,C;)
perfect matching for all v E V ( H ) .
Theorem 2.1.4. [29] Let G = ( Kz, F, , . . . , Fk) and assume that each Fi is hypomatchable.
Then there exists a polynomial-time algorithm to find a maximum site G-packing of arbitrary
input gmph H .
This theorem will not be proven here. The proof given in [29] uses an extension of
Lemma 2.1.1. In order to extend Lemma 2.1.1. we need t o extend t he augmenting configu-
rations to the case G = (Icz , FIT. . . , Fk). Let G and G" be 6-packings and let an alternating
path refer t o a G, G'-alternating path in H whose edges are alternately G-edges and C"-edges.
T h e augmenting configurations are then as follows.
1. An augmenting path is an atternating path between two vertices exposed in G.
2. An augmenting tail is an alternating pa th between a n exposed vertex of G and a
vertex covered in G by some G - Fi (not by a G-edge).
3. An augmenting kite is an even length alternating path between an exposed vertex u of
G and a vertex v of some Fi such t h a t the Fi and the alternating path intersect only
at v. Also, there exist $(I V ( F i ) I - 1) vertex-disjoint alternating paths between pairs
of vertices of Fi - u such t ha t each path begins and ends with a G-edge. An example
of such a kite is given in Figure 2.8.
In this Figure, the f ((v(F;) ( - 1) vertex-disjoint alternating pa ths are the paths 1,3,
and 2,6,5,4.
T h e following proposition is analogous to Lemma 2.1.1.
C'H-4 PTER 2. SURVEY OF THE GENERAL GRA PH P..\CI\'ILVG' PROBL EAf 20
Proposition 2.1.2. 1291 Lei G satisfy the clssumptions of Theomm 2.1.4. The G-packing
G of a graph H is of mazimum size if and only if it has no augmenting p l h . tail? or kite
in [ I .
There is a corollary t o Theorem 2.1 .-l which considers a possibly infinite family of cliques.
Corollary 2.1.2. [27. 29, 17, 361 If C {K1, K 2 , K 3 , . . .) (G not necessan'ly finite) and
Ii-, E 6 or E G, then there is a polynomial-time algorithm for the maximum site G - packing problem.
P m f : Clearly K r E G is trivial and G = 1'-2 is solvable by a polynomial
masimum matching algorithm. Then consider the case where K2 is in G a n d
all o ther cliques in G are even. All elements of G have perfect matchings and
so a maximum size G-packing is found by searching for a maximum size (K2)- packing.
If K 2 is in 6 and Kt, t > 1, is t he smallest odd clique in G, then a n y G-packing
h a s a ( K 2 , Kt)-factor since o d d cliques can be covered with a copy of and
some number of copies of K 2 , a n d any even clique can be covered with copies of
IC2. Therefore: i t is sufficient to find a maximum size { K 2 , Kt)-packing which
by Theorem 2.1.4 can be done in polynomial time. 0
2.2 Families of Complete Bipartite Graphs
Recall t h a t a complete bipartite g raph having a partition of size one a n d t h e other of size
k is called a star a n d is denoted S k . T h e sets {Sly S2,. . . , Sk) and { S r , Sz, . . .) a r e called
sequential star sets.
T h e search for a polynomial algorithm t o find a maximum {S,,. . . , Sk)-packing s temmed
from a n observation of Las Vergnas, given below, concerning (1, k)-factors. A ( 1 , k)-factor
is a spanning subgraph with all degrees between 1 a n d k.
Proposition 2.2.1. [47] A graph has a {S1,. . . ,Sk)-factor if and only if it has a ( 1 , k)-
factor.
Proof: Given a graph H which has no isolated vertices, consider the following
O(I El) algorithm.
C'H-4PTER 2. SURVEY OF THE GENERAL GR4 PH PACf\'INC PROBLEhl 21
Algorithm 2.2.1. To jnd a {S1. . . . . Sk)-/actor o r {Sl, Sz , . . . )-factor of H .
I . For each edge trv f H , delete uv if both u and v haw degree greater lhan
one.
2. Update the degrees of u a n d v.
If all degrees of the vertices of H a r e between 1 and k, then this algorithm
will find an {Sl? . . . . Sk)-factor. T h u s a (1. k)-factor found by any algorithm
for finding (1, k)-factors ['LZ, 441, can be modified t o give an (SI, . . . , Sk)-factor.
Since the ( 1 , k)-factor problem is polynomial, the {S l , . . . , Sk)-factor problem is
also polynomial.
Note tha t if the algorithm is applied t o a n arbitrary graph H , then it will find
a maximum sequential s t a r packing, t h a t is an {Sly Sz, . . . )-packing. Therefore.
the {Sly Sz, . . . )-packing problem is polynomial.
I t is proven in [28, 301 t h a t for any other families G of complete bipartite graphs, the
G-packing problem is NP-complete o r harder. We will not prove this here.
Proposition 2.2.2. [28, 301 A graph II has an {Sl! S2 , . . .)-factor if a n d only if it has no
isolated vertices.
Proof: This foIlows from Proposition 2.2.1. Since each vertex must have degree
at least 1, no vertex can b e isolated. O
T h e augmenting configurations referred to in Theorem 2.2.1 beiow a re shown in Fig-
ure 2.9. T h e number o r letter beside each vertex denotes the degree of t h a t vertex in the
packing. Brackets with a n asterisk around a par t of the configuration denote t h a t t h a t part
of the configuration may be repeated q times, q > 0. As before, bold edges denote edges
covered by the packing and circled vertices denote vertices uncovered in the packing.
When we augment these configurations, we obtain the configurations of Figure 2.10.
Theorem 2.2.1. [2R, 30) Let k 2 2 . The size of a mazimum {Sly.. .,Sk}-packing o / H is
equal to the minimum, over all T C_ V ( H ) , of n + k - IT1 - ir where n = Ik'(H)[ a n d i~
denotes the number of isolated vertices of H \ T.
CHAPTER 2. SCIRVE)' OF THE GENERAL GRAPH P-~CKIIVG PROBLEM 22
(ii) ( i i i )
Figure 2.9: Augmenting s t a r configurations
(9 (i i ) ( i i i )
Figure 2.10: Augmented s tar configurations
Proof: If T is a subset of V ( H ) , then the largest number of vertices in H \ T
which a r e not exposed in H occurs if each vertex in T is the cent re vertex
of an Sk a n d all t h e o the r vertices of t h e Sk a r e in H \ T . T h i s number is
k ITI. Therefore, t h e number of exposed vertices in the packing G of H is
at least t h e number of vertices which a r e exposed in H a n d th i s number is
i~ - k - ITl. Therefore, the number of covered vertices in t h e packing G is at
most n - (iT - k ITI) = n + k . IT1 - i~ for any 2' S V ( H ) . Now, to prove this
equality we define a packing G a n d a se t T C V ( H ) such t h a t G covers exactly
n + k - IT1 - i~ vertices of H .
Let G b e any {Sl , . . . , Sk}-packing of H which admits no augment ing configu-
ration. Let U denote t h e se t of exposed vertices in G. Let t h e sets T a n d PV be
defined as follows:
If degcv = k and u is adjacent to a vertex of U U W in H , then v f T.
CH.4PTER 2. SURVEY OF T H E GENERAL GRAPH P.AC'f\'lrVG PROBLEA9l
If w is adjacent t o a vertex of T in G, then w E 1.2'.
Since G admi t s no augment ing configuration, each neighbour of a ver tes u E U
must have degree k with respect t o G' and so be a n element on T . .;\lso. any
w E bV must have degree 1 in G a n d be reached from some u by a pa th o n which
the degrees of the vertices a l t e rna te between k a n d 1 with the vertes adjacent t o
u having degree k and the v e r t e s preceding w having degree k. Since t h e vertes
preceding w has degree k a n d is adjacent t o w € W , i t must be in T a n d so each
neighbour of a w f W is a n element of T. Consequently, all vertices of Cr u W
must be isolated in H \ T, a n d s o i~ 2 IU( + IW(. T h i s means t h a t G covers
n - lUl > n+lWI - i~ = n + k - IT1 -iT vertices. O
Corollary 2.2.1. [28, 301 An {S1, . . . , Sk)-packing G of H is maximum if and only if it
admils no augmenting configurntion.
P m f : Clearly, if G a d m i t s a n augmenting configuration then G is not maxi-
mum. If G does no t admit a n augmenting configuration, then by the proof of
Theorem 2.2.1, there exists a set T C V ( H ) such t h a t G covers n + k - IT1 - iT vertices of H and s o is a maximum packing. 0
Another corollary t o Theorem 2.2.1, given beIow, was proven independently by Amahashi
a n d Kano f6] and by Las Vergnas [47]. I t can also be found in [3, 91.
Corollary 2.2.2. ['La, 301 Let k 2 2 . A graph H admils an {Sly . . . , Sk)-factor ij and only
if it does not have a set T of vertices whose deletion results in more than k [TI isolated
ue rt ices.
P m f : ['28, 301 B y Theorem 2.2.1, t h e maximum size of a packing of H is t h e
minimum of n + k 17'1 - iT over all T. If there is a set T such t h a t iT > k - [TI,
then n + k - IT1 - i~ < n. Therefore, the maximum size of a packing will be n if
and only if it 5 A:. 17'1 for each T & V ( H ) . CI
2.3 Families of Paths
Having considered families of complete graphs and of corn plete bipart i te graphs, we now
shift o u r at tention to families of paths. Recall t h a t in th is thesis, Pk represents t h e pa th
with k edges.
CWAPTER 2. SURVEY' OF THE GEXERAL GRAPH P.4C,'i<IArG PROBLEM
T h e maximum G-packing problem where G is the pa th on o n e edge, P I , is equivalent t o
the niasirnum matching problem which is well known t o be polynomial.
The Family {Pz}
I t follows from Theorem 2.1.1 t h a t the {P2}-packing problem is NP-comple te . We now give
a proof of this using t h e local replacement technique.
Theorem 2.3.1. The ( P2) -factor problem is JVP-complete.
Proof: This problem is in AfP since if we a re given a potential {P2)-factor of
a graph H as a certificate, we can verify tha t all vertices of H a r e covered by
exactly one P2 in polynomial time. We will reduce t h e N P - c o m p l e t e problem
3DM t o the {P2}-factor problem to establish t h e NP-completeness. Consider
the gadget of Figure 2.1 1.
Figure 2.11: Gadget for t h e {Pz)-factor problem
Construction: Recall t h a t 3DM considers a se t 7- = { ( t l , t z l t3) l t l E W,t2 E
X, t3 E Y } , some subset of W x X x Y where W, X, Y = (1,2,. . ., g ) . We will
construct a graph G as follows: Let W U X u Y be the vertices of a graph. For
each triple ( t l , t 2 , t3) of T, identify t h e connector vertices, a, 6, a n d c, of a copy
of the gadget with t h e corresponding vertices t l , t2 and t3 of t h e graph. T h e
edges of these copies of the gadget a re in the edge set of t h e g raph a n d t h e
interior vertices a r e in t h e vertex set. Call the resalting g raph G.
Claim 2.3.1. The set T has a 3-dimensional matching if and only if G has a
( P2) -Jacior.
C,'HA PTER 2. SURVE)' OF THE GENERAL GRA PM PACKllVC PROBLELLI 2.5
Proof: Suppose 7- has a 8-dimensional matching. Then for chosen
triples ( t tz , t 3 ) , Figure 2.12 shows t h e { P2)-factor of the copy of t h e
gadget having connector vertices t l = a. t2 - - 6 and tJ = C.
F igure 2.12: A { Pz)-factor containing t h e connector vertices
For unchosen triples, t h a t is when 11, t 2 , a n d t S a re each covered by
some o t h e r triple, t h e copy of t h e gadget having connector vertices
t l = a , t 2 = b a n d t S = C, has a {P2)-factor as shown in Figure 2.13.
Figure 2.13: A {P2)-factor excluding t h e connector vertices
Thus , the re is always a {P2)-factor of G when there is a 3-dimensional
matching of T.
Suppose G has a {P2)-factor. Consider a copy of the gadget. If vertex
a is covered b y t h e P2 21x20, then a' can be covered by choosing afb'yz
o r a'b'd. If we choose a'b'y2 then there is n o way t o cover yl. Thus , we
must choose a'b'c' a n d so b will b e covered by y1y2b and c by 2 1 ~ 2 ~ as
in Figure 2.12. In this case, t h e triple (a, 6, c ) is in the 3-dimensional
matching.
If instead a is covered in some o the r way, then X I must be covered
by t h e P2 x1x2a'. Thus, 6' can b e covered by yly2bf, by btc'z2, o r by
CH-4PTER 2. SURVEY OF THE GENERAL GRAPH PACKlNC' PROBLEM 26
1~~6'd . In the last case, we are unable t o cover vertes yl . Choosing
6'8~~ wotild leave the vertex uncovered in the packing and so we
must choose ~~3~6'- Therefore, b and c will be covered by Pzs other
t h a n y1y26 and 21z2c as in Figure 2.13. In this case, t h e triple (a, b, c )
is not chosen as a triple of the 3-dimensional matching.
T h e resulting set of chosen triples 7- = {(a, b, c ) ) is 3-dimensional
matching. CI
Since t h e iVP-complete problem 3DM reduces t o t h e {P2)-factor problem, the
{ Pz)-factor problem is also NP-complete. 0
An interesting class of input graphs for which the {P2)-packing problem has been studied
is the class of squam graphs. Let i E ( 1 , . . ., n} and j E (1,. . . , m). Consider the grid graph
defined by V = (1, . . . , n) x (1, . . . , m) with n m vertices such t h a t for i, k E (1, . . . , n) a n d
j, 1 E ( I , . . ., m ) , vertex (i, j) is adjacent t o vertex (k, 1 ) if and only if i = A: and I j - 11 = 1
or li - kl = 1 a n d j = 1. A square gmph is a subgraph C of the grid graph with the property
tha t G is connected and each edge of G is contained in some C; of G. An example of a
square graph with a (&)-packing is given in Figure 2.14.
Figure 2.14: A {P2)-Packing of a Square Graph
Some results concerning square graphs and their {P2)-factors a re given in the next
theorem.
Theorem 2.3.2. [5] Let G be a square gmph on p vertices.
(I) If p 0 (mod 3 ) , G has a {P2) -factor.
(11) If p r 1 ( m o d 3) and v is an arbitrary vertex ojdegree 2 in G, G\ v has a {Pz)-factor.
The Family { P I , P2)
Results on the packing problem for t h e fami[?; { P I , P2) can be found in [3, 1. 2. 51.
.An interesting result on (P , , Pz)-factors. Corollary 2.3.1, concerns triangle graphs. Con-
sider a triangulated grid. -4 connected subgraph of this grid in which every vertex is con-
tained in a IT3, is called a t rimgle graph.
We have the foIlowing corollary concerning { P I , Pz)-factors of triangIe graphs. Such a
factor is shown in Figure 2.15. This corollary was proven by -4kiyama and Kano.
Figure 2.15: A {P t , P2)-Factor of a Triangle Graph
Corollary 2.3.1. [.5] Every triangle graph T has a {PI . P2)-factor.
Proof: Let S be a vertex subset of T. A ver tes x of T is incident with at most
six vertices. Thus, t h e maximum number of vertices incident with z which can
be isotated in T \ S is four, as shown in Figure 2.16.
Figure 2.16: -Maximum isolation of vertices incident with x
Also, each isolated vertex of T \ S must be incident with at Ieast two vertices of
S. Therefore,
CHAPTER 2. SURVEY-OF THE G E N E R A L GRAPH P.4CKlNC: PROBLEM 28
where. i(T \ S ) is the number of isolated vertices in T \ S.
Thus , by Corollary 2.2.2, with k = 2. a triangIe graph must admit a { P , , P2)- factor.
2.4 Families of Cycles
T h e two-factor problem, which is t h e 6-factor problem where G = {C3,C4,. . . ) ? is poly-
nomial p24. In addition t o this, t h e g raph packing problem is polynomial for the family
= {C4,C;, . . .), see [31] and for t h e family G = {G,C5,C6,. . -) [7]. T h e graph pack-
ing problem for t h e family G = {G, G,. . -) is expected t o be polynomial, see [31], bu t
this problem remains open. For all o t h e r finite families of cycles, the packing problem is
N P - c o m p l e t e r29. 311. T h e JVP-completeness proofs for the packing problem with families
G = {C6,C7, . . .), = {C3tC4,C6,C;T...), a n d = {C4,C6,C?, ...) can b e found in [18],
a n d with t h e family G = {C3, C5, C;, . . . ) c a n b e found in [ G I .
Two-Factors
A (C3, Cl, . . . )-factor, of a graph H is a subgraph F of H for which V ( F ) = V ( H ) and
every vertex has degree two. Thus, a two-factor of H consists of disjoint cycles t h a t cover
1'- (H) .
Theorem 2.4.1. [46] A gmph G = (V, E) admi t s a {C3, C4,. . . )-factor if and only if
where IC is the se t of vertices ofdegree at most 1 in G\A, and x is the number of cornponenls
of G \ A which contain an odd number of vertices from I<.
2.5 Families of Mixed Graphs
T h e packing problem where the farniIy G consists of I<;! along with some other g raph was
studied by Cornut5jols, Hartvigsen a n d Pulleyblank [IT], by Hell a n d Kirkpatrick r28, 36, 29,
301 a n d by Loebl a n d Poljak [39, 401. Loebl a n d Poljak have completely characterized the
complexity of deciding whether a perfect packing exists for such families.
CH.4 PTER 2. SURVEY OF THE GENER4 L G R A P H PACIiliVG PROBLEM 29
Some early work on the {K2 , F)-packing problem studied the problem for F = P2
[ 3 ? 28. 471 and for F hypomatchable [IT. 16, 291. In both cases the packing problem is poly-
nomial. These results gave hope for other polynoniial results. T h e main polynomial result
of this section, Theorem 2.5.1, concerns the class of graphs F which are perfectly matchable,
hypomatchable o r propellers. A propeller is a graph obtained from a hypomatchable graph
I+- by adding two new vertices, c? t h e centre. and r , t h e root, the edge cr and some edge(s)
from c to a nonempty subset of IY.
Theorem 2.5.1. [41, 401 The {K2, F)-packing ~roblern is polynornially solvable /or all
gmphs F which are either perfectly matchable, hypomatchable, or a propeller. The pmb-
fern is NP-complete for all other gmphs F .
I t is clear t h a t if all graphs F are perfectly matchable, then the problem is equivalent to
t h e maximum matching problem. The theorem was proven in [li] for graphs F which are
hypomatchable and in [41] for graphs F which a r e propellers. These two proofs will not be
presented here.
2.6 Applications
-4s we have mentioned in Chapter 1, matching theory has many real-world applications.
This theory has been used in scheduling, assignment, transportation, network flow. shortest
path , travelling salesman and Chinese postman problems.
A specific application for the graph packing problem is the area of examination scheclul-
ing. In [27], Hell and Kirkpatrick consider such a problem. T h e problem of minimizing the
number of exam periods and avoiding conflicts such as students taking two exams a t a time,
is a well-studied problem and is NP-complete. T h e problem studied in [27] is t h a t of g r o u p
ing exam periods into sets of k periods. Consider a set of courses which have already been
separated into examination periods. T h e k periods represent k examination days, and the
exams in a given period are organized within t h a t day's schedule. T h e goal is t o minimize
the second order conflicts, such as a student being required t o sit two exams on t h e same
day o r with only a given amount of time between exams. This problem can be formulated
in a graph theoretic way.
CHAPTER 2. SURVEY OF THE GEXER.4L GRAPH P.4CIIiING PROBLEM 30
Inslance: -4 graph G. a positive integer n and w a weight function o n t h e graph K, such tha t LJ : E(Ii,) + LO, 00).
Question: Are there d disjoint subgraphs C1, G2, . . . . C'd of such t h a t
each Gi is isomorphic to G'.
each vertex of Kn is in some Gi, and
d
2 Cee.q~.) w ( e ) is minimized? i= 1
In this definition of t h e problem. the graph G, with k vertices, corresponds t o the
definition of t h e second order conflicts. T h e graph I<, represent the n examination periods
and we can assume tha t n = kd. T h e weight function, w ( e ) , where e = ( u , v ) , corresponds t o
t h e number of students who need t o write a n exam in both period u and period u. Consider
the example used in [27]. If G = K3, then k = 3 and the solution GI , . . . , Gd, corresponds
t o a partition of the examination periods into d days with 3 periods per day. T h e number
of occurences of a student being required t o write two consecutive exams on the same day
is minimized.
A s we can see from the question stated for this problem, we are looking for a G-factor
of I<,. Notice t h a t in the formulation discussed above, i t is assumed t h a t t h e scheduler has
eliminated first order conflicts.
Chapter 3
Directed Graph Packings
Study of t h e graph packing problem for undirected graphs resulted in many polynomial
algorithms as well a s some JVP-completeness results. We now move o u r focus to the graph
packing problem for directed graphs.
.4fter introducing t h e definitions and notation t o be used in th is chapter, we will look
a t the directed graph packing problem for directed paths, cycles a n d s tars .
3.1 Definitions and Notation
Recall from Chapter 1 t h a t a directed graph o r digraph H = ( V ( H ) , A ( N ) ) is a graph with
the se t V ( H ) of vertices and t h e s e t A ( H ) of arcs. Recall t h a t a n a r c a6 is oriented from
u t o 6. If we are not interested in the direction, or d o not know the direction on a given
arc, we will refer to i t simply as an edge. A directed G-packing of a digraph H is a set of
pairwise vertex-disjoint subgraphs of H each of which is isomorphic to some digraph in the
family G. A vertex of H is covered by the directed G-packing if i t belongs t o a subgraph of
t h e G-packing. If t h e directed G-packing has size IV(H)Iy then it is a directed G-factor.
T h e directed G-factor problem can be s ta ted as follows:
Given t h e family of directed graphs 6,
Directed G-factor: Instance: A digraph H .
Question: Does H admit a directed G-factor, t h a t is, is there a packing
of H using digraphs from t h e family G such that the size of
the packing is IV(H)I?
C'H.4 PTER 3. DIRECTED CRA PH PACl\'lhrC;S :3 2
For the remainder of this chapter. the terms G-packing a n d 6-factor will refer to directed
G-packing and directed G-factor unless otherwise stated.
-AS we have mentioned, we will s tudy this problem for directed paths. cycles and stars.
.-\ direcled lmth on k edges, denoted f i . is a graph with vertex se t {vl , . . . , a k + l ] and arcs
{ ~ ; v ; + ~ li E {I, . . . , L)}. A directed cycle on k edges, denoted G, is a graph with vertices
{ v I , . . . , uk} a n d arcs {vl up, u2U3, - - . , uk-1 vk, ukvI }. Finally, a directed star on A- edges,
denoted s;, is a complete bipart i te graph with partitions {u) and { v l . . . ., v k } and arcs
{uv,Ii € { I , . . ., k)}. T h e underlying g raph of a directed graph H is the g raph H' o n t h e same vertex set as
N. T h e edge uv is in E(H') if uv E A ( H ) or vu E A ( H ) o r both.
3.2 Packing with Directed Paths
3.2.1 One-Element Families
The Family 6 = {PI}
Theorem 3.2.1. The maximum {fi}-packing problem is in P.
Proof: For a digraph H, consider the underlying undirected graph H'. Then
the {a}-packing problem for H becomes the maximum matching problem for
H', which is known to be polynomial. Therefore, if we find a maximum matching
of t h e g raph H', then by choosing the arcs of H corresponding t o the matched
edges of H', we obta in a maximum {fi}-packing of H in polynomial time. O
The Family C = {fi} Theorem 3.2.2. The {e} - factor problem is NP-complete.
P m f : T h e proof of this theorem is analogous t o t h e proof of Theorem 2.3.1. If
we have a {p2}-factor as a certificate, we can verify i t s correctness in polynomial
time. Therefore, t h e {&}-factor problem is in NP. We will now reduce DDM t o t h e (6 )-factor problem. Consider the gadget of Figure 3.1.
Recall t h a t an instance of 3DM is a set of triples T = {(t t 2 , t3)lt E W, t Z E X, t s E Y ) which is some subset of W x X x Y (W, X , Y = {1 ,2 , . . ., n}). We
CH.A PTER 3. DI R EC'TED GR.4 PH P..IC.'f<flVCS
Figure 3.1: Gadget for t h e { e l - f a c t o r problem
will construct a graph G such t h a t 7 has a 3-dimensional matching if and only
if G admits a {GI-factor.
Construction: Let the vertex se t of a graph consist of one vertex for each element
of M' U ,Y u Y . For each triple of T, a d d a copy of the gadget to t h e graph by
identifying connector vertex a with vertex t l , connector vertex b with t2, and
connector vertes c with t 3 . T h e interior vertices and the edges of t h e copy of
t h e gadget become vertices and edges of the graph. Call this graph G .
Claim 3.2.1. The set 7- has a 3-dimensional matching if and only if G has a
{ e l - f a c t o r -
P m f : Suppose ';r has a 3-dimensional matching. Then for triples
( t L , t 2 , t 3 ) in the matching we have t h e {p2)-factor of the copy of the
gadget, having connector vertices t l = a, ta = b, and t3 = c, shown in
Figure 3.2.
Figure 3.2: A {p2)-factor containing the connector vertices
C'N.4 PTER 3. DIRECTED GRA PH PACIiINGS
For triples not in the matching, tha t is when t I , t 2 , and t 3 are each
covered by some other triple, the correspondirig copy of the gadget in
G has a { e l - f a c t o r as in Figure 3.3.
Figure 3.3: A (6) - fac tor excluding t he connector vertices
Thus, there is always a {fi }-factor of G when there is a Mirnensional
matching of 7.
Suppose G has a (6)-factor . Consider a copy of the gadget. The
vertes x1 can be covered by choosing t he directed path xlx2a o r the
directed path xlx2at. If i t is covered by ~ 1 x 2 ~ ~ then in order to cover
a', atbtc' must be in the packing. This in tu rn forces yly2b and zlzzc
to be in the packing. Thus, if one of the directed paths incident with
a connector vertex is in the packing, then the connector vertices of
the gadget a r e covered as in Figure 3.2. In this case, (a, b, c) is chosen
as one of t he triples in the 3-dimensional matching. If X I had been
covered by t h e directed path x1z2at , then the only way t o cover b'
would have been to choose the directed path yl y2bt and this would
force t l z2c ' to b e in the packing as well. T h u s if the directed path of a
gadget incident with a connector vertex is not covered by the packing,
then none of t he connector vertices of t he gadget will be covered by
the packing. In this case, the triple (a, 6, c) is not in the 3-dimensional
matching.
T h e resulting set of chosen triples is a 3-dimensional matching. a
Thus, the (6 ) - f ac to r problem is NP-complete.
C'H.4 PTER 3, DIRECTED G R4 PN P.~CI<IIVGS 3.5
T h e gadget used for t h e ( 6 ) - f a c t o r problem can b e used for o t h e r directed paths of 1 1 1 length two by altering the a r c directions in the paths x1x2n. xlxza', a G c , yly2b. 91y2b'.
z1z2c, and z I z 2 c 1 accordingly. O n e such nlodification of the gadget is used in the NP- completeness of the $;-packing problem on page 64.
The main result of this chapter concerns the graph packing problem with the family {fi: f3} and will be given in Section 3.3.
T h e graph factor problem with the family {PI, &} is actually t h e same as the {el- factor problem since any copy of f?! can be perfectly packed with two copies of the graph
. Therefore, the {pl, 4 ) - f a c t o r problem is polynomial. A similar argument shows t h a t
all {ply 6 )-factor problems where j is odd a re polynomial.
We will now consider t h e { f i , El}-factor problem. O u r approach is based upon work by
Loebl and Poljak in [dl]. We will consider the {fi, &}-factor problem with a restriction on
t h e vertex se t of the inpu t graph. If this restricted problem is JVP-complete, and we will
show t h a t it is, then t h e {PI, q ) - f a c t o r problem is also JVP-complete.
In the restricted { P I , fi)-factor problem stated below, the vertex se t V ( G ) of input
graph G is made u p of t h e disjoint union of a set D and a se t .4 of vertices. T h e se t D is
independent, which means t h a t for all u, u E Dl uu 4 E ( G ) .
restricted { f i , fi)-factor: Inslanee: Digraph G with V(G) = D(G) u A(G) , such t h a t D is independent and 2101 =
3 1 4 Question: Does G a d m i t a {PI, fi}-factor?
Proposition 3.2.1. A digraph G with V ( G ) = D(G) U .4(G)! such that D is independent
and 21DI = 31AI has a {fi, fi}-/actor i j and only if G admits a { e l - j a c t o r F such (hat for
each of the k fis 01 F, vertices UI, u3, and US a m vertices of D and vertices u2 and u4 are
vertices o j A.
Proof: Consider a digraph G with V ( G ) = D(G) u .4(G), such t h a t D is inde-
pendent and 2101 = 31AI. If there is a {fi}-factor F of G, with t h e requirement
s ta ted in the proposition, then there is a {fi ,f i)-factor a n d there a r e 3k vertices
in D and 2k in A.
CHA PTER .3. DIRECTED GR-4 PH PACI\'INC;S 36
Suppose t h a t we have a {fi. fi}-factor. We will prove by counting that the
vertices in D covered by the factor. a r e covered by f i ' s meeting the requirement.
Suppose t h a t the factor uses i 6 ' s such t h a t vertices vl , Q, and us cover vertices
of D a n d vertices "2, a n d "4 cover vertices of A: j fi's which cover two vertices
of D; n 6 's which cover one vertex of D; p fi's which cover no vertices of D: 1
6 ' s such t h a t one vertex covers a vertex of .4 and the other a vertex of D; a n d rn
P1's such t h a t both vertices cover vertices of A. Since 2101 = 4-41, we have the
2 ( R i + 2 j + n + l ) = 3 ( 2 i + 3 j + 4 n + 5 y + 1 + 2 m )
following: 6 i + 4 j + ' L n + 2 1 = 6 i + 9 j + 1 2 n + I.5p+31+6nz
0 = . 5 j + 1 0 n + 1 . 5 p + 1 + 6 m
Since j, n, p, 1 a n d m are non negative integers. the only solution is t h a t j = 4
n = p = 1 = m = 0 and so we only use P4's obeying the requirement in t h e
factor a n d so we have a ( 6 ) - f a c t o r of G. 0
Consider t h e colouring of a 6 shown in Figure 3.4.
Figure 3.4: Colouring of a fi used in C O L { ~ ~ }
We define C O L { ~ ~ } as follows:
COL-{fi}: Instance: A directed g r a p h G a n d a colouring 4~ : V ( G ) + (0, I} of t h e vertices of G.
Question: Does G admit a {GI-factor which is faithful to t h e
colouring?
Theorem 3.2.3. The C O L - { f i } -/aclor problem is AfP-complele.
P-f: We will reduce 3DM to t h e COL-{el-factor problem. Consider t h e
gadget of Figure 3.5 with connector vertex b and connector edges a and c.
Consider a n instance of 3DM for which t h e s e t s W and Y consist of q arcs o n
two vertices each, and t h e se t X consists of 9 vertices.
C'H.4 PTER .3. DIRECTED G R4 PI-I P-4CKINGS
Figure 3.5: Gadget for the COL-{el-factor problem
Construction: Consider the vertex set V ( W ) u X u V(Y) and the arc set con-
taining the arcs from sets IY and Y. For each triple (.!, t',. t2, Isti) of T, add
a copy of the gadget t o the graph by identifying connector arc alas with tit{, connector vertex b with t2 , and connector arc clc2 with t&. The interior arc of
these copies of the gadget make u p the arc set of the graph. All interior vertices
of the gadgets a re added to the vertex set. Call the resulting graph G'.
Claim 3.2.2. The set T has a 3-dimensional matching if and only i j G has a
factor.
If 7' has a 3-dimensional matching, then for triples in the matching,
we have the COL-{el-factor of the corresponding copy of the gadget
shown in Figure 3.6.
Figure 3.6: A ~oL-{f i ) - fac tor of the gadget including the connectors
For triples not in the matching, the corresponding copy of the gadget
has a COL-(6 }-factor as shown in Figure 3.7.
C'H.4 PTER .?. DIRECTED C R-4 PH P.4Cl\'INCS
Figure 3.7: -4 ~ ~ ~ { f i ) - f a c t o r of t h e gadget excluding the connectors
Thus, for any 3-dimensional matching of 7, we have a COL-{el- factor of G.
Suppose tha t we have a COL-{fi j-factor of G. Consider the vertex I
of a given copy of the gadget. If I is covered by a f i which contains the
vertes t but not the vertes w , then LL: cannot be covered. Therefore, x
must be covered either by the f i between w and 2. or by the f i from
b t o z and then t o y. Suppose z is covered by the f i from w t o z.
Then y can only be covered if the p4 from z t o y is chosen. In this
case, none of the connectors are chosen and so the triple (a, 6 , c) is not
in the 3-dimensional matching of 7.
If I is covered by the fi; from b t o z t o y, then w can only be covered
by choosing the f i from w t o a for the packing and then r can only
be covered by choosing the f i from r t o c. In this case, all connectors
will be covered and so we choose t he triple (a, b, c) to be in the 3-
dimensional matching of T.
This is a 3-dimensional matching of T since it is generated from a
{e l - fac tor .
The COL-{el-factor problem is NP-complete.
Now tha t we have shown the ~oL-{fi]-factor problem t o be NP-complete, we can prove
the NP-completeness of the 16, a}-factor problem by reduction of the COL-{GI-factor
problem.
Theorem 3.2.4. The Restricted {pl, F..)-/act~r problem is NP-complete.
C1H.4PTER 3. DIRECTED GRAPH PAC'IilNGS
Pmof: We will reduce COL-{fi} to t h e restricted {fi, PI}-factor problem.
Suppose G is a n instance of COL-{el. W e will c rea te a n instance G' of t h e
restricted {PI, l?,}-factor problem in t h e following way: Firstly, remove edges
having bo th end vertices coloured 1 since these edges a r e not used in t h e { f i . G } - fac tor problem. All o ther edges become t h e edge se t of G'. All vertices with
colour 1 make up the independent set D and those with colour 0 make u p t h e se t
A of VfG'). If we have a Y E S i n s t a n c e o f COL-{fi}. the se t D must have size
3P a n d the se t A must have size 2k, a n d all fi's will have vertices from D a n d
A as shown in Figure 3.8. Since G a d m i t s a COL-{fi j-factor, G' must a d m i t a
{ e l - f a c t o r . Th i s implies t h a t G' a d m i t s a {PI, l?4)-factor.
Figure 3.8: Placement of f i
Now, suppose G' is a YESins tance o f t h e restricted {fi, 6 ) - f a c t o r problem.
T h e n by Proposition 3.2.1, there is a {&}-factor F of G'. If we colour vertices
t . 1 , u~ a m d us of each f i of F with colour 1 a n d vertices 02 and "4 with colour
0 , t h e n we have a factor of G'. 0
Corollary 3.2.1. The {Fl, l?4)-factor problem is A'P-complele.
P-f: Any instance of the restricted {fi , f i}-factor problem is also a n instance
o f t h e {PI, fi}-factor problem since b o t h probiems ask t he same question. 0
Following a similar approach to t h a t used for t h e {fi, fi}-factor problem, i t c a n b e
shown t h a t all (6, e}- fac to r problems where j is even and greater o r equal to 4, a r e
N P - c o m p l e t e . Th i s again follows from ideas used in [41].
Infinite Families of Paths
Stemming from t h e gadget for the {fi}-packing problem, as well as from gadgets const ructed
for t h e I&}-, {fi, A}-, {fi, a}-, and ( f i , fi}-pa~king problems individually, we have been
C H A P T E R 3. DIRECTED GR4PH P4CKINGS
able t o const ruct a gadget which can be used t o prove the JVP-completeness of families of
directed pa ths f i with k 2 2. Interestingly, o u r gadget works for all such families except the
family { fi,&}. However, t h e different approach t a b n in [12] allows this case to be treated
and i t is found t o be NP-comple te as well. O u r gadget is constructecl in the following way:
Construction: Consider a family G of directed paths. Let pk be the shortest pa th in
t h a t family. Let 8 E G and $! G, where 2k - 1 2 I 1 k + 2. Then we construct the
gadget shown in Figure 3.9.
Figure 3.9: Gadget for pa th factor problems
T h e connector b is a directed p a t h whose length is 2k - 1 - 1 . T h e inequality 21; - 1 2 1
ensures t h a t t h e connector b has at least one vertex. T h e inequality 1 > k + 2 ensures t h a t 4
there is at least one vertex between t h e Pks x1 . . . xk+l a n d yl . . . yk+l. This gadget does
not work for t h e {fi, f i}-pcking problem since t h e inequality for this case gives 3 2 3 > 4.
We have the following theorem:
*
Theorem 3.2.5. The G-factor problem is .UP-complete for G { f i , . . . , 4, . . . }, where
2 / ; - 1 1 1 > k + 2 , ! : 2 2 , 8 ~ ~ o n d P ; + ~ $ ~ .
Proof: We will reduce 3DM to this G-factor problem. Consider t h e gadget of
Figure 3.9 with connector vertices a a n d c and connector b a pa th of length
2k - I - 1. Consider t h e 3-dimensional matching problem with W and Y, two
se t s having q points, and X , a set having q elements, each isomorphic t o a path
CHAPTER .3. DIRECTED GRAPH PACKINGS 41
of length 2k - I - 1. For t h i s proof, we consider triples o f instances of 3DM to
consist of t from W , t g from 1' and a p a t h t 2 , . . . . t 2 2k-l from X.
We cons t ruc t t h e g raph G as before on t h e vertex set WU-Y UY. For each triple
of 7, we a d d t h e arcs a n d inter ior vertices of a copy of t h e gadget t o t he g raph - by associat ing a with t l , b wi th t h e PZk-I-tt t Z , , . . . , t22k-r, a n d c with t s .
Claim 3.2.3. 7 has a 3-dimensional matching if and only if G has a G-factor.
P m f : Suppose t h a t 'T has a 3-dimensional matching. Then for
triples in t h e matching, we have a G-factor of t h e corresponding copies
of t h e gadge t s as shown in Figure 3.10.
Figure 3.10: A G-Factor including t h e connectors
For those triples no t in t h e matching, we have a G-factor of t h e corre-
sponding copies of t h e gadge t s as shown in Figure 3.11.
Figure 3.11: A G-Factor excluding t h e connectors
CHAPTER .3. DIRECTED GRAPH P,.\CfiliVCS
Thus, if we have a 3-dimensional matchir~g of T, then we have a G- factor of the graph G.
Suppose t h a t we have a 6-factor of C'. Consider one of t h e copies of the
gadget in G. Vertex zl can either be covered by t h e f i 1 1 . . . I ~ I ~ + I , - by the 4 X I . . . r l + l , o r by the Pk X I . . . t k a . Suppose t h a t it is covered
by the first. Then we canno t use the connector 6. Therefore, the - vertices t k+2 , . . . ,21+1 canno t be covered by a. f i o r by a f l since the
path X k + 2 . - . x l + , has length less than k. T h u s suppose tha t x l is
covered by the 6 X I . . .21+1. Then in order t o cover yk+l, we must - choose the Pk yl . . . yk+l. In this case, none of t h e connectors a r e in
the G-factor a n d so t h e triple (a , b, c ) is not pa r t of t h e 3-dimensional
matching of T. 4
Suppose t h a t xl is covered by the Pk x l . . . xka. Then since xk+l . . . zr+l
and xk+l . . . xl+l yk+l a r e b o t h pa ths of length less than k, the oniy - way t o cover zk+l is t o choose t h e Pk bl . . . b2k-1xk+l . . . xl+lyk+l- This
then forces us t o choose the 6 yl . . .ykc in order t o cover t h e vertices
yl . . . yk. In this case, all connectors a re covered a n d so ( a , b, c) is par t
of t h e 3-dimensional matching of T.
Thus, if we have a G-factor of G, we have a 3-dimensional matching
of T. 0
T h e F-factor problem for G C {a .. -8 - . . I , 2k - 1 2 1 2 k + 2 , k 2 2, fi E G, GI 4 4 , is NP-complete . 0
3.3 Main Theorem
This theorem makes use of augmenting configurations. An augmenting conjigurntion of a
graph H with respect t o a packing P of H is a special subgraph of H such t h a t the number of
vertices covered by P in t h e subgraph can be increased, o r augmented, by removing certain
edges from P and adding o the r edges to P , o r simply by adding edges t o P. RecaIl t h a t an
a r c ub is a directed edge beginning at ver tes a a n d ending in vertex 6. If we d o not know
the direction of a n arc o r if t h e direction is not important we will simply refer t o i t as the
edge ab . A walk is a sequence voe lv le2 . . . e k v k of vertices and edges such t h a t e; = v;- 1 v;
CH.4PTER 3. DIRECTED GRAPH PACI\'ING'S
for all i. If no edges of the sequence a re repeated. then we have a tmil. We will simply give
t h e vertices of t h e trail in this proof. There are directed edges between consecutive vertices
of t h e trail. A11 of our augmenting configurations a re trails. Note t h a t a vertes may be
repeated in a trail. In this case, the augmenting configuration is set'j-intersecting,
Theorem 3.3.1 s ta tes our main result: a {fi, fiJpacking is maximum if and only if there
a re n o augmenting configurations. A polynomial time algorithm for the {PI , &)-packing
problem can b e found in [12]. This algorithm is inspired by Hall's theorem and the resulting
algorithm for matchings in bipartite graphs. In the remainder o f this section, t h e term
packing will be used t o refer to a 16, fi)-packing.
In this chapter, a n alternating tmil is a trail UO, VO, U,, v1, - . . vk-1, uk with a n even
number of edges, where uo is t h e s t a r t vertex, a n exposed v e r t e s of P . For each i f
{O? 1,. . . , k - 1) we have t h e following:
a u;v; is a n edge of H which is not in t h e packing P .
a q u , + ~ is a n edge of the packing P.
a T h e edges uivi and v;u;+l a re either both oriented towards the vertex tl; o r both
oriented away from the vertes u;.
a u;+l is incident with exactly one arc of P a n d v, is incident with one o r two arcs of P.
Note t h a t if u; is incident with two arcs of P , then one of those arcs is directed towards
v; a n d one is directed away from v;. Also, n o vertex u; can a p p e a r twice in a n alternating
trail.
An augmenting configuration is a n alternating trail of one of t h e following types:
1 . T y p e 1: A trail UO, VO, U I , v1 . . . , Z L ~ , 2, where uo, vo, u1, . . . , uk is a n alternating trail T
a n d z is a n exposed vertex of P.
2. T y p e 2: A trail uo, v0, ul, ~1 . . . , uk, Z, where uoT t~o, ul?. . -, uk is a n alternating trail T
a n d
(a) r = o;, 0 5 i 5 k - 1. Then ukviu; is a p2 but is no t in P and v;u;+l in P is not
p a r t of a p2 of P, o r
(b) z = u;, 0 5 i 5 k - 1. Then U k U i V i is a p2 but is not in P and v i u i + ~ in P is no t
pa r t of a f32 of P.
C'M.4 PTER -3. DIRECTED GR4 Pff P.-\C'I\'fNC;S
3. Type 3: A trail uo, UO, U I , vl -. ., uk, y. z . where rro. ~'0. u l . . . .: u k is a n alternating trail.
u k y is not in P and yz is in P. -
(a) ukyz is a P2. or
(b) z = v i 7 0 <_ i 5 k - I. Then yviui+l is a f3 in P.
4. Type 4: A trail uo, vo, u l , v l . . . , uk, x, y, Z, where uo. uo, u l , . . . ? uk is an alternating
trail. u p is not in P and zyr is a fi of P.
T h e augmenting configurations of type 1: 3a, and 3 are not self-intersecting. Recall t ha t
this means no vertex is encountered more t han once in the trail. Augmenting configurations
of type 2a, 2b and 3b are self-intersecting. Esamples of each type of augmenting configura-
tion are given in Figure 3.12. In this figure, dot ted lines are edges which are in the packing
P and solid lines are edges which are not in P . Circled vertices are exposed in P . In all of these types of augmenting configurations, .uo is a n exposed vertex which we call
the s t a r t component, any sequence of vertices u;viui+l is called a middle component and x,
y and 2 are end components. These components are represented in Figure 3.13.
C'H.4 PTER 3. DIRECTED GRA PH P.4 C,'i\'liVGS
Configuration
Type 1
Example
"1 "I 1 U 2 X Y z Type 4
----------).
Figure 3.12: Examples of augmenting configurations for the { f i , p2]-packing problem
C'H-4PTER -3. DIRECTED GRAPH PACKINCS
Start components
Middle components
End components
* denotes that this vertex may have been previously encountered
Figure 3.13: Augmenting configuration components for the {fi. f i ~ - ~ a c k i n ~ problem
C'M.4 PTER .3. DIRECTED GRAPH PACKIXGS
Now we will look at how augmenting configurations a re augmented. Augmentecl config-
urations for each esample of Figure 3-12 are given in Figure 3.14.
Configuration Augmented example
Figure 3.14: Examples of augmented configurations for t h e {PI, &}-packing problem
Consider the type 1 configuration. T o augment, remove edges v;u;+l, i = 0, . . . k - 1 from
P and add edges u p ; , i = 0, . . . , k - 1 and ukz to P . For t h e type 2a and 2b configurations.
remove edges v;u;+l, i = 0, . . . k - 1 from P and add edges u;v;, i = 0 , . . ., k - 1 a n d ukr t o P . For a type 3a configuration, remove edges viui+l, i = 0, . . .k - 1 from P and add edges uivi,
i = 0 , . . . , k - 1 , uky to P a n d keep edge yz in P. To augment the type 3b configuration,
remove edges viui+l, i = 0 , . . .k - 1 and yz from P and add edges uiui, i = 0 , . . ., k - 1 and
CHAPTER 3. DIRECTED GRAPH P.4C'KfNG'S
trk. t o P. FinaIly. t o augment type 4 configurations. remove edges t 7 ,u ;+ l . i = 0 , . . .A- - 1
and xy from P. add edges u iv ; , i = 0 , . , .. k - 1 and ukx t o P and keep edge g z in P. tn Theorem 3.3.1, we will use the notation G \ ab, where ab E E(G) t o represent the
subgraph of G having the same vertex set as G and with edge set E(G\ ab) = E ( G ) \ a l .
Theorem 3.3.1. A {PI. 6) -pucking P o/ o directed gmph G is of maximum size if and
only if there exists no augmenting configurntion.
Proof: Suppose G contains an augmenting configuration with respect to P.
Then clearly by augmenting we will increase the size of P and so i t could not
have been of maximum size-
Suppose that P is not of maximum size. Let P' be a larger {fi, fi)-packing of
G. We will prove t ha t P must admi t an augmenting configuration.
We prove the following statement by induction on IE(G)I! the number of edges
in the graph G:
For all k , if G has k more vertices covered by P' than by P , then there are k
vertices which start an augmenting conjgunrtion in G with respect to P .
Consider the case I E(G) I = 0. In this case, k = 0 and so we have 0 augmenting
configurations and the statement holds.
Suppose tha t the statement holds for all k for any graph with fewer than J E(G')l
edges. Suppose t ha t G has {PI, & ~ - ~ a c k i n ~ s P and P' such tha t there are
k 2 1 more vertices covered by P' than by P. We will use the terminology
P-edge and P'-edge to denote t h a t a n edge is a 6 of P , or of P', respectively.
Similarly P-path and PI-path. will denote a p2 which is in the packing P , or
of P' respectively. Note we can assume E(G) = P u PI. If not, consider the
spanning subgraph of G with edge set P U P ' and apply our statement. T h e k
vertices which begin augmenting configurations in this spanning subgraph also
begin augmenting configurations in G. Let v be a vertex covered only by P'.
Then we have the following three cases t o consider:
1. T h e vertex v is covered by a Pt-edge vw.
2. T h e vertex v is covered by a P'-path vwu.
3. T h e vertex v is covered by a P'-path wlvwz.
CHAPTER 3. Dl RECTED GRAPH PAC,'IiI!VGS
We will now t rea t each case with i t s accompanying subcases. All these proofs
follow a similar format. We will look a t a subgraph G\ab of G ancl we will prove
t h a t k vertices s t a r t augmenting configurations of P restricted t o G'\ab and these
a re either preserved o r can be modified t o produce augmenting configurations
in G. Note t h a t in the following proofs. t h e term augmenting configuration will
always refer t o a n augmenting configuration with respect t o the packing P .
1. Suppose vertex u is covered by a Pr-edge cw.
( a ) Suppose u: is also esposed in P. Consider t h e graph G \ v w a n d t h e
packings P a n d P' restricted t o G \ uw. In this graph, there a re k - 2
vertices covered by P' a n d no t by P a n d so by our s ta tement , there a r e
k - 2 vertices which begin augment ing configurations in G \ vw. Since
v a n d w a re exposed with respect t o bo th P and P' in G \ urn, they
canno t appear in any augmenting configurations. Consequently, none
of t h e k - 2 vertices beginning augmenting configurations a re affected
by t h e edge uw and so all augmenting configurations a r e preserved
in G . Clearly, both v a n d u; begin augmenting configurations in G.
namely vw and wv, and so we have k vertices which begin augmenting
configurations in G.
(b) Suppose w is covered by a P-edge wu a n d wu @ P'. Consider t h e graph
G \ wu. In this graph, P' covers k + 2 more vertices t h a n P. Vertices v
a n d w begin augmenting configurations, v w a n d wv, in G \ wu. Since
neither u nor w a re incident with any P-edges or P - p a t h s in G \ wu,
these a r e the only two augment ing configurations in which they a p
pear. Therefore, there a r e k o the r vertices which begin augmenting
configurations and we need only consider those augmenting configura-
t ions which contain u since all augmenting configurations which d o not
involve u a re augmenting configurations of P in G as well. T h e ex-
posed vertex u must ei ther s t a r t o r end a n augmenting configuration.
Suppose there is a n augmenting configuration of type 1 U ~ V O U ~ . . . ukz, where z = u. If ukuw is a 4, then uovo. . . U ~ U W is a n augmenting
configuration of type 3a in G . Otherwise, ukuw is a middle component
a n d then uovo.. . u ~ u w v is a n augmenting configuration of type 1 in
C'MAPTER 3. DIRECTED GRAPH P-4CKINGS
G. At this point, we know t h a t any ver tes uo # u which s t a r t s a n
augmenting configuration of P in C \ tuu which does not contain u o r
which ends a t u also s t a r t s a n augmenting configuration in G. We also
know t h a t there a r e at least k - 1 such vertices. If there a r e El., then
the proof is complete. If not , then the k t h s t a r t vertes mus t be u.
Suppose u begins a n augmenting configuration U V ~ U ~ . . . u k E t E = z,
y r , o r zyz , in G \ mu. If vwu is a 6, then uwu is a n augment ing
configuration of t y p e 3a s ta r t ing a t v in C. Otherwise, uwuvo . . . ukE
is a n augmenting configuration in G. This gives the k t h vertex which
begins a n augment ing configuration in G .
(c) Suppose t h a t w is o n a P - p a t h wuu'. Consider the g raph G \ wu. In
this graph, P' covers k + 1 more vertices than P. As before, vw a n d
wv are augmenting configurations so we will turn our a t t en t ion t o the
remaining k - 1 vertices? o the r than u and w t which begin augment ing
configurations in G \ wu. Since u and u' a re covered by a P-edge
in G \ mu, neither c a n s t a r t a configuration. Consider a n augment ing
configuration which begins a t a ver tes uo. Suppose tha t uu' is a P-edge
in this configuration. Note t h a t u a n d u' cannot appear on a P-edge
o the r than uu' in a n augment ing configuration since uu' belongs t o the
P-pa th wuu' in G a n d this would imply t h a t one of them was incident
with three P-edges in G o r was on a pa th of P of length three. Since
t h e argument for such a configuration, with uu' a P-edge in C\ wu and
pa r t of a P - p a t h in G, will b e used frequently throughout t h e proof
of Theorem 3.3.1, we wilI prove t h e following lemma t o which we will
refer throughout t h e remainder of the proof.
Lemma 3.3.1. Let G be o graph, P a {PI, 6 ) - p a c k i n g of G and let a6
be a P-edge of some P-path. If G \ a6 has an augmenling configumtion
of P beginning at uo, uo # a, 6, then either there is an augmenting con-
figuration beginning at uo in G o r there is an augmenting configumtion
of type 2a wilh a = z and b not in the configumtion.
Pfoof: Suppose a is t h e middle vertex of the P - p a t h a'ab,
otherwise reverse t h e roles of a a n d 6 . In the graph G \ ab, if b
is a n exposed vertex in an augmenting configuration beginning
CN-4PTER 3. DIRECTED GR.4PH P'4C.'IiINGS
at a vertex uo, which does not contain a' o r a. then b = 2. a n d
so in G we woiild have a n augmenting configuration uo . . .baa'
of type 4.
If ver tes a is covered by some P-edge a'a in some augmenting
configuration uouortl . . . vk-1 uk E in G \ ab, where E = z, yx o r
xyz, then either b is in t h e configuration o r i t is not. If 6 is
in t h e configuration, then b = z a n d we have an augmenting
configl~ration of P in G of type 3b with a = v,. and a' = uj+r.
ff b is not in the configuration, then a is a u;, 0 < i 5 I; o r a v; ,
0 5 i < k. If a = u; in any type of configuration, then a' = u;-1
a n d in G we have a type 4 ending a'ab t o the configuration.
I t remains to consider the case where a = vi- If a = u; and in
t h e trail from uo t o a no vertex is revisited in the augmenting
configuration, tha t is, we have configurations of type 1, 3a o r 4
o r we a re in the pa r t of configurations 2a, 2b and 3b between uo
a n d t h e revisited vertex, then a' = u;+l. and in G the P-edge
a6 is pa r t of a middle of this augment ing configuration and so
t h e augmenting configuration is preserved in G'.
Now suppose a = v; and t h e vertex u, o r uj, j 5 i is revisited in
t h e augmenting configuration. T h e possible configurations a r e
those of types 2a, 2b and 3b, First , consider an augmenting
configuration of type 2a. Either i = j o r i > j. If a = v; ,
a n d i = j, then we have the second result of the lemma. If
a = v;, and i > j, then in G this augment ing configuration will
be preserved, the P-edge a b will jus t be par t of a middle.
Suppose that the atlgmenting configuration uouo .. . ukz is of
t y p e 2b. Then vertex uj is revisited a n d for a = v;, we have
2' > j and a' = ui+l in t h e al ternating trail uovo.. .uk. In G we can find a new augmenting configuration beginning at uo
in t h e following way. Travel along t h e alternating trail from
uo to uj. Instead of continuing to uj , follow the edge ujuk.
If ujurvk-1 is a 4, then we have encountered a n augmenting
configuration uovo . . . U j U k V k - 1 of t y p e 3a. Otherwise, we have
C'HA PTER 3. DIRECTED GRAPH PACYiINGS
a middle u, ukuk-1- In this case, if vk-1 is the middle vertex of
a P-path . then we have a n ending of type 4. If not, we have
a middle -ujukvx.,l and we look a t the nest edge covered in
P , u~,~uc;-L o n the aIternating trail. Again, we will have a n
ending of type 3a, o r of type 4 o r we will have a middle. If
we have a middle, continue t o t h e next edge. We u 4 l find a n
ending since t h e vertex a = v; of the edge aa' is t h e middle
vertex of the P - p a t h a'ab.
Finally, suppose t h a t a n augmenting configuration of type 3b
is found in G \ ab. Vertex a could never equal uj, where uj is
t h e revisited vertex, since this would mean t h a t a was incident
with three P-edges in G. Thus, a = v; and a' = u;+ l , i > j ,
a n d in G this augmenting configuration is preserved since a6
will just be par t of a middle of t h e augmenting configuration.
0
We now return t o case l c where w is on a P-pa th wuu'. Note tha t
for a configuration of type 2a in G \ wu, if u = v; = z , then in G
we have the augmenting configuration u o ~ . . -u ,u t . . . UWU. Augment
this configuration as follows: -411 arcs of the augmenting configuration
which were covered in P' become covered in P and uncovered in P ' ,
and all arcs of t h e augmenting configuration which were covered in P become covered in P' and uncovered in P.
.411 augmenting configurations of P found in G \ wu are preserved o r
can b e modified t o produce new augmenting configurations, beginning
at t h e same vertex uo, in G, as shown in Lemma 3.3.1. Therefore,
there a re k - I vertices which begin augmenting configurations in G
and which d o not begin at t. o r w. T h e augmenting configuration vwuu'
gives us k vertices which begin augmenting configurations in G.
(d) Suppose w is the middle vertex of a P - p a t h U'WU, oriented from u' to
u. Either vw and u'w form a middle component o r vw and uw form a
middle component. Say i t is vw and u'w, otherwise reverse the name
of u a n d u'. Consider the graph G \ U'W. Since the vertex u' is exposed
in G \ u'w, P' covers k + 1 more vertices than P . Therefore there are
C'HA PTER 3. DIRECTED G'RA PH PACl\'lhrGS
a t least k vertices other than u' which s t a r t augmenting configurations
in C \ ulw. By Lemma 3.3.1 ail a r e preserved in G except possibly a
type 2a configuration with w = vjjz and u = u j + l . However. such a
configuration cannot exist since this would mean t ha t w was incident
with three PI-edges in G.
2. Suppose vertex v is covered by a P'-path vwu.
(a) Suppose w is exposed in P and consider the graph G' \ v u;. The packing
P' covers k - I more vertices than P in G \ v,w and so there are k - 1
vertices which begin augmenting configurations in G \ cw and all of
these configurations are also configurations in G. T h e reason for this
is t ha t adding the PI-edge vw to any augmenting configuration cannot
destroy the configuration since w is exposed in C \ vw. Since vw is
an augmenting configuration of G beginning a t v, we have k vertices
which begin augmenting configurations in G.
(b) Suppose tha t w and u are covered in P by (i) the P-edge wu, or (ii)
a P-path wuu', o r (iii) a P-path u'wu. Examples of these cases are
shown in Figure 3.15. Note tha t in these diagrams, the edge wu is in
both P and P'.
(0 (ii) ( i i i )
Figure 3.1.5: Example diagrams for case '2b
For cases ( i) and (ii), consider the graph G \ vw. In this graph, P' covers k - 1 more vertices than P. Since no vertices covered in P have
become exposed in P through the restriction t o G \vw, any augmenting
configuration beginning a t uo involving zu, u o r u' in G \ vw is still a n
augmenting configuration beginning at uo in G since adding P'-edges
cannot destroy augmenting configurations.
In case (i) , vwu is an augmenting configuration of type 3a in G, and in
CHA PTER :3. DIRECTED GrW PH PACKINGS
case (ii), uwurr' is an augmenting configuration of type 4 in G and so in
both cases, there are k vertices which begin augmenting configurations
in G.
For case (iii), consider the graph G \ wu'. In this graph. 13' covers
k + 1 more vertices than P. T h e vertex u' is esposed with respect t o
P in G' \ wu'. If ut begins an augmenting configuration, then in G,
this augmenting configuration will be destroyed. Therefore, there a r e
k other vertices which begin augmenting configurations in G \ u:uf. T h e
rest of this proof follows from Lemma 3.3.1. Note t ha t we cannot have
an augmenting configuration of type 2a with w = r and u = uj+l since
this would imply tha t w had degree three with respect t o P' in G. Thus
there a re 12 vertices which begin augmenting configurations in G.
(c) Suppose t ha t wu is not in P . Then w can be covered by (i) a P-edge
aw, or (ii) a P-path a'aw, or (iii) a P -pa th awu'. Examples of these
cases a re shown in Figure 3.16.
(i) ( i i) ( i i i)
Figure 3.16: Example diagrams for case 2c
In cases (i) and (ii), consider the graph G \ aw. In case (iii), if vwa' is a
6, consider G \ aw. Otherwise, vwa is a fi and then consider G \ a t u .
In case (i), P' covers k + 2 more vertices than P. In cases (ii) and (iii),
Pr covers k + 1 more vertices than P. In cases (i) and (ii), vw and
wv are augmenting configurations and so there a re k and k - I other
vertices respectively which begin augmenting configurations in G \ aw.
Consider case (i). If an augmenting configuration contains a, then i t
either begins o r ends in a. If i t begins at a, then we can construct a n
augmenting configuration beginning at v as we did in case l b . Suppose
a n augmenting configuration begins a t some uo # a and ends a t a
CHAPTER .3. DIRECTED GRAPH P-4CIilhrGS
- with the edge ukn. If uka and aw form a P2. then uo . . . ukaw is a n
augmenting configuration of type 3a in G'. Otherwise. a wu provides a n
ending of type 1 for the augmenting configuration in G.
Suppose a n augmenting configuration ends at ver tes w. Then it must
be a n augmenting configuration uovo .. . urn of t y p e 1. If uwa is a I%'2.
then uua is a type 3a configuration in G. Otherwise, uwa must form a
middle. In th is case, we consider vertex a which either begins an aug-
menting configuration o r does not. If a begins a n augmenting configura-
tion a . . . uk E, then consider uot.0 . . . uw and a . . . uk E . where a # uo. If
no vertex of a .. .ukE appears in uovo.. . uw. then uovo.. . u w a . . .ukE
is a n augmenting configuration of G. If a ver tes is repeated, then i t is
a vi, 0 5 i < k, o r a u;, 0 < i 5 k. If vertex u; is t h e first vertex of
a .. . ukE which is also in uovo . . . uw, then uo . . . wwa . . . v; is an aug-
menting configuration of type 2 a in G provided t h a t v is not the middle
vertex of a P - p a t h. If vi is the middle ver tes of a P - p a t h, then consider
uo . - . t'; - . . wa . . . v; . . . ukE in G . First augment between v; and E by
switching along the pa th v; . . - uk and augmenting t h e ending E. Then
uo . . .u; . . - wa.. .u; is a n augmenting configuration of type 2a in C. If
a vertex u; is t h e first vertex of a .. . ukE which is also in uovo . . .uw, then uo . . . uwa . . . u; is a n augmenting configuration of type 2b in G.
Now consider uovo. . . uw and a . . . uk E. where a = uo. T h e n there is
one augmenting configuration which begins at n a n d one which ends at
w. This is simply a n augmenting configuration which begins a t a and
ends at w. Then we can construct an augmenting configuration as in l b
and we preserve o u r k vertices which s t a r t augmenting configurations.
Now suppose we have a n augmenting configuration uovo .. . uw of type
I , uwa is a middle, and a does not begin a n augmenting configuration
in G \ aw. We claim t h a t i t is possible t o modify so t h a t only one
augmenting configuration ends a t w. T o see this, suppose tha t two
augmenting configurations end at w, one beginning at a vertex uo and
the other at a vertex c. Both must end with t h e Pf-edge U W . NOW
either u is t h e first vertex common t o both augmenting configurations
o r i t is not. If i t is, then t h e augmenting configuration beginning at uo
C'H.4 PTER 3. DIRECTED GRAPH P=IC'f\'IXC;S
has a P-edge 1 1 u and the augnienting configuration beginning at c has
a P-edge lzu. T h e edges 1 , u and 12u must form a p2 and so one of the
augmenting configurations, say the one beginning at c can be modified
t o an augmenting configuration c . . . lzul 1 of t ype 4. Then there is just
one augmenting configuration ending a t w.
If u is not the first intersection of the two augmenting configurations,
then the first intersection is at some vertex vi, 0 5 i < k, or u i?
0 < i 5 k- Note that if vo is the first intersection, the uouoc is a P'- 4
path and so either uovoul o r maul is a P2 and hence an augmenting
configuration of type 3a which does not end a t w. If the augmenting
configurations first intersect at a u;, then either l lui is a P-edge of
u o . . . l l u ; . . . w and 12ui is a P-edge of c . . . l z u i . . .w, o r ll.ui is a P- edge and f2u; is a P'-edge. In the first case, l lu i fz is a P-pa th and
so one augmenting configuration, say the one beginning a t c can be
modified t o become the augmenting configuration c.. .12.ull of type 4 ,
and the augmenting configuration beginning at uo and ending at .w is
preserved. In the second case, we have lzuit.; which is a Pf-path, not
a middle and so the augmenting configuration beginning a t c is not
legitimate.
If the augmenting configurations first intersect at some vertex .v;, then
either l l v ; and 12vi are PI-edges o r else llui is a P'-edge and 12vi is
a P-edge. In the first case, either l l ~ i ~ i + l or 1 2 ~ i ~ i + l is a F$ and
so the corresponding augmenting configuration is actuaiiy of type 3a
with ending f l ~ i ~ i + l or 12~iu;+ l and so there is only one augmenting d
configuration ending at w. In the second case, f 2 v ; ~ i + 1 must be a P2
and so c . . .12u;~i+l is an augmenting configuration of type 4 and only
one augmenting configuration ends at w.
Therefore, only one augmenting configuration ends at w. !f uwa is
a fi, then this augmenting configuration acquires the ending uwa of
type 3a in G. Otherwise, this augmenting configuration is destroyed
by the addition of aw. However, in this case, vwa is a n augmenting
configuration beginning a t vertex u and so we still have k vertices which
begin augmenting configurations in G and we are done.
C'H.4 PTER 3. DIRECTED GR4 PN P.4CIiIArGS
Case (ii) follows from Lemma 3.3.1. In t h e case of an augmenting
configuration of type 'La with a = u, = z , a n d a' = r r , + , . the aug-
menting configuration is destroyed in C' but vtcaa' is a n augmenting
configuration of G' and so we still have k vertices beginning augmenting
configurations.
In case (iii)? if we a r e considering C \ aw, then vwa' is a n augmenting
configuration of C . Otherwise, vwa is a n augmenting configuration of
G\af w . In either case, there a re k other vertices which s t a r t augmenting
configurations in G \ aw, o r G \ a'w. Case (iii) then follows from
Lemma 3.3.1. An augmenting configuration of type 'La not involving
v with w = uj = r? and a' = uj+l (or a = u,+l if we have G \ d w ) ,
cannot exist since w cannot b e incident with three PI-edges. If t he
augmenting configuration does s t a r t a t v , then i t is destroyed in G and
we use t h e augmenting configuration vwa' (or vwa) instead.
3. Suppose vertex o is covered by a Pf-path w,vwz, oriented from w l t o Z C ~ .
If ei ther wl o r w2 is exposed in P , proceed as in case 2a.
(a) Suppose t h a t w l a l is a P-edge and t h a t t h e a r c w2a2 is covered by (i)
t h e P-edge wza2, o r (ii) by the P - p a t h w2a2a',. These cases a r e shown
in Figure 3.17.
Figure 3.17: Example diagrams for case 3a
Consider t h e graph G \ wza2. In case (i) , there a r e k + 2 more vertices
covered by P' than by P. In case (ii), this number is k+ 1. In bo th cases,
vwz and w2u a re augmenting configurations and so there a re k other
vertices in (i) and k - 1 in (ii) which s t a r t augmenting configurations
C'NA PTER 3. DIRECTED G R.4 PH P-4 CId!VCS
in G\ w2a2. T h e remainder of case (i) is analogous t o case Ib and case
(ii) is analogous t o case lc. since w l and a l are no t affected by the
removal of the edge ~ 2 ~ 2 .
If the direction of wlvwz is reversed, then case (i) is t h e same as case
3b (i) and case (ii) is unaffected.
(b) Finally, suppose vwza2 and uwlal are middles. Then (i) 62W2 is a P- edge, or (ii) naw2ai is a P-path . These cases a re shown in Figure 3.18.
Figure 3.18: Example diagrams for case 3b
Consider the graph G \ w2a2. T h e number of vertices covered by P' and not by P is k + 2 in case (i) and I; + 1 in case (ii). If we disregard
the augmenting configurations beginning at v and w2 in case (i) and
beginning a t v in case (ii), there remain A; other vertices which begin
augmenting configurations in both case (i) and case (ii).
Consider case (i). T h e vertex a2 is exposed and so i t may s t a r t o r
end augmenting configurations. Suppose a n augmenting configuration
s ta r t s a t a2. Then vw2az s ta r t s this configuration in G. If we have a n +
augmenting configuration uo . . . ukaz of type 1 , then if ur,a2w2 is a P2, we have a n augmenting configuration of type 3a in G. Otherwise, a2w2v
provides a new ending of type 1 in G. An augmenting configuration
cannot end a t w2 since .w2 is the end vertex of t h e P'-path wlvw2 and
so no other P'-edge can contain w.
For case (ii), see Lemma 3.3.1. Again, we cannot have a n augmenting
configuration of type 2 a with a; = uj+l and w2 = vj = 2, since w2 is
incident with t h e P'-path wl vwz in G.
CH.4PTER 3. DIRECTED GR-4PH PAC'IiIiVGS
If the direction of wl r w z is reversed. then case (i) is the s ame as case 3a
(i). For case (ii) , reverse t he roles of a2 and a: and the case is identical.
In a11 cases, if there a re k vertices of G' esposecl in P, then there a re k vertices
which begin augmenting configurations of P in G'. This completes the proof of
our theorem. D
3.4 Packing with Directed Cycles
In this section, we consider packing with one-element families of directed cycles. We will
not consider the { ~ i } - ~ a c k i n ~ problem since it is polynomial time solvable analogously t o
t he {fi }-packing problem of Theorem 3.2.1.
The Family B = {G}
Theorem 3.4.1. The {c}- fac tor problem is JVP-complete.
Proof: This problem is in t h e class JVP since given a {c3}-factor. we can check
tha t all vertices are covered by one and only one d j in polynomial time. In
order t o prove the {c3}-factor problem NP-complete, we will reduce 3DM to
the {C3}-factor problem.
Consider the gadget of Figure 3.19 having connector vertices a', a2, and a3. We
construct a graph G in the following way. Let the vertex set of G be WU S U Y.
For each triple of 7- = {( t l , t z , t3 ) l t l E W.t2 E X, tS E Y), add a copy of t he
gadget to the graph by identifying a1 with t l , a2 with t2 and as with t 3 .
Claim 3.4.1. The set T has a .?-dimensional matching i j and only il G has a
{C3} -factor.
P m f : Suppose T has a 3-dimensional matching. Then for each triple
in the matching we have t h e {GI- fac tor of Figure 3.20.
For each triple not in t h e matching, we have the {c3}-factor of Fig-
ure 3.21.
Clearly there is always a {GI- fac tor of G when there is s 3-dimensional
matching of T.
C'H.4 PTER 3. DIRECTED GRAPH PA CKlNGS
Figure 3.19: Gadget for the {G}-factor problem
Figure 3.20: A {G)-factor containing the connector vertices
Suppose we have a {GI-factor of G. This part of the proof is similar
t o t ha t of Theorem 2.1.1, the undirected case of this problem, with
the exception t ha t here we have connectors a l , a;! and as in place of
a, b and c. The reader can see t ha t just as in Theorem 2.1.1, if a
connector, say al, is covered by a lx2xl , then the only way to cover
23 is to have 233313 in the {c3)-factor and this in turn necessitates
having yla2y2 and rla3z2 in the {C3}-factor. In this case, we choose
the triple ( t l , t2, t3 ) of T t o be in the 3-dimensional matching. If a1
is covered by a 6 3 other than zlal tz, then in order t o cover tl and
2 2 , we must choose 2122Z3 . Eventually, we find t h a t a2 and a3 must
C'I-I.4 PTER 3. DIRECTED GRAPH P.~CI<IIVGS
Figure 3.21: A {e3}-factor excluding t h e connector vertices
also be covered by c3's o t h e r than yla2y2 a n d z ' a 3 z ~ in this case,
we d o n o t choose t h e triple (t t z , t S ) of T t o be in the 3-dimensional
matching. Th i s produces a 3-dimensional matching of T. a
Thus, 3DM a {c3)-factor problem and so t h e {c;)-factor problem is ,LrP-
complete. CI
The Family E = {ck) Theorem 3.4.2. Let k 2 3 be an integer. The {&}-factor problem is NP-complete.
Proof: As was the case for t h e {c3}-factor problem. the (ck}-factor problem
is in NP. We will give the construction of the gadge t using t h e case c4 as
our example. T h e proof of this theorem uses reduction of t h e k-dimensional
matching problem.
Construction: To construct a gadget for this problem, we begin with a directed
having vertices yl,. . .,yk. To each of the vertices of this central cycle we
attach G ' s which a re vertex-disjoint from each other . To each of these k c7#'s we a t t ach a connector vertex a;, i = 1.. .k, using two directed edges. These
edges a re incident with the two vertices on a n edge with a vertex y;, i = 1, . . . , k.
T h e addition of these connector vertices produces k new C;s. A n example of
the gadget for k = 4 is given in Figure 3.22.
C1H.4PTER 3. DIRECTED GRAPH PACKIIVG'S
Figure 3.22: Gadget for t h e {c4)-factor problem
As in o u r o ther proofs, we construct a graph G with vertex set Wl U M/rzU- -U Wk. For each k-tuple of T = ( ( t l , . . . , t k ) l t l E CVI,. . . , tk E M/i,} we add a copy of
the gadget t o G by identifying connector vertex a1 with vertex t 1 , a;! with t 2 e t
cetera.
Now, we claim t h a t 7 has a k-dimensional matching if and only if G has a {ek)-
factor. Once again, consider the case k = 4. If T has a +dimensional matching,
then for a 4-tuple in the matching, the corresponding copy of t h e gadget in G
has the ( G I - f a c t o r of Figure 3.23.
If a C t u p l e is not in the matching then we have the {c4}-factor of Figure 3.24
of t h a t copy of t h e gadget in G.
Thus G has a {GI-factor.
We continue t h e argument as in the proof of Theorem 3.4.1 t o prove tha t kDM cc {GI- fac to r problem and so the {Ck)-factor problem is NP-complete.
3.5 Packing with Directed Stars
The 15;)-factor problem is polynomial as shown in Theorem 3.2.1.
T h e {$}-factor problem is NP-complete. T h i s can b e proven by reduction of 3DM.
T h e gadget used is a modification of the gadget for t h e ( 6 ) - f a c t o r problem and is shown
in Figure 3.25. T h e proof of NP-completeness is similar to t h a t of Theorem 2.3.1.
C'H.4 PTER 3. DIR EC'TED G RA PI1 PAC~l<ltVGS
Figure 3.23: A {c4}-factor containing the connector vertices
We will give a full proof of the NP-completeness of t h e {GI-factor problem and then
show how t o const ruct a gadget which can be used t o prove t h e .1VP-completeness of any
{,$;.-factor problem through reduction of the k + 1-dimensional matching problem. T h e
NP-completeness of t h e {.S%}-factor problem will not be shown as it is similar t o t h a t of
t h e {c3}-factor problem with t h e obvious alterations and renaming.
The Family G = S3
Theorem 3.5.1. The {GI-factor problem is NP-complete.
This problem is in the class JVP. We will reduce 4DM to t h e {&}-factor problem
t o prove NP-completeness. T h e gadget for this problem is given in Figure 3.26.
We will cons t ruc t t h e graph G on vertex set Wl u W2 u W3 U W4. For each &tuple
( t t , t 2 , t3 , t 4 ) of T, add a copy of t h e gadget to G by identifying a1 with t l , an
with t 2 , ag with t3 , a n d a4 with t4 .
Claim 3.5.1. 7 has o 4-dimensional matching if and only if G has an {GI - factor.
Proof: Suppose there is a 4-dimensional matching of 7. Then for
4-tuples in t h e matching, we have t h e {&}-factor of Figure 3.27 for
the copy of gadget associated with t h a t 4-tuple in G.
C'H.4 PTER 3. DIRECTED G R-\ PN P.4 CI\'INC;S
Figure 3.21: A {c4}-factor excluding t h e connector vertices
Figure 3.25: Gadget for the {&}-factor problem
For those 4-tuples which are not in t h e matching, we have t h e factor
of Figure 3.28 of the corresponding copy of the gadget.
Suppose G has an {&}-factor. Consider a copy of of t h e gadget. If
a1 is covered by t h e 6 centred at t l and not containing y, then in
order to cover y, t h e & with vertices a, a, 1 2 and ZJ must be in the
{$}-factor. This would produce the factor shown in Figure 3.27. In
this case, we choose t h e 4-tuple ( t l , t s , tS, t4) to be in the 4-dimensional
matching of T.
If a, is no t covered by t h e & centred at X I and not containing y, then
a similar argument proves t h a t t h e factor of Figure 3.27 is t h e only
one possible and we d o not choose t h e $-tuple ( t l , t2 , t3 , l4 ) t o be in
CH.4 PTER 3. DIRECTED GR.4 PH P.4CKIfNC;S
Figu r adget for the {%)-lacto
the 4-dimensional matching of T.
This gives a 4-dimensional matching of T. 0
Therefore, the {$}-factor problem is JW-complete. O
The Family 4 = $
As for the {ck)-factor problem, we will show how to construct a gadget for any k t o be
used in proving the JW-completeness of the {s~J-factor problem. Once again, we will
demonstrate this construction for the case k = 4.
Theorem 3.5.2. The {.$;I - factor problem is NP-complete.
Construction: Consider an 5 centred at a vertex y and having arcs ( y , zi),
i = 2 . . k + 1 Attach a n & centred a t a vertex zl t o y with the arc zl y. - Similarly,attach a n Sk centred at a vertex xi, i = (2 , . . . , k + 1) t o each vertex
z; with the arc ziz;. Choose one of the unnamed vertices of each s tar centred at
xi and name it a;, i = 1, . . . , k + 1. These are the k + 1 connector vertices of the
gadget. When k = 4, this construction produces the gadget of Figure 3.29.
To prove Theorem 3.5.2, reduce the k + 1-dimensional matching problem t o t h e
{&}-factor problem as before. The rest of the proof is left t o the reader. 0
CHAPTER 3. DIRECTED GRAPH R4CKINGS
Figure 3-27: An {&}-factor containing the connector vertices
It"
Figure 3.28: An {s )-factor excluding the connector vertices
CHA PTER .3. DIRECTED GRAPH P.4CKlNGS
a4 a3
Figure 3.29: Gadget for the (.!&}-factor problem
Chapter 4
Conclusion
The purpose of this thesis was t o s tudy the graph packing problem for directed graphs.
Much work had been done on the undirected graph packing problem and so we felt tha t
s tudy of t h e directed cases might yield interesting results, O u r work was based upon both
matching theory and the work of many mathematicians on various undirected graph packing
problems. As a result, the second chapter was dedicated to giving the reader a n introduction
to some relevant resuIts on t h e undirected packing problem. In addition, we endeavoured t o
demystify t o some extent certain areas of JVP-completeness theory pertinent t o our work.
W e t rus t t h a t after reading this thesis, the reader has some understanding of the realm of
NP-completeness and of the usefulness of this theory in studying and proving the various
degrees of difficulty of problems. O u r NP-completeness proof technique of choice is the
local replacement technique which employs the use of gadgets as demonstrated extensively
in Chap te r 3. Other methods of proof exist and t h e reader is encouraged t o peruse [24].
We d o not claim tha t Chapter 2 provides a complete survey of all work done o n undi-
rected graph packings. T h e graph packing problem has been approached and defined in so
many different ways tha t a complete survey would produce many volumes of work. We d o
hope however, t h a t Chapter 2 provides a good representation of the work done on t h e type
of undirected graph packing problems which we chose t o study in the directed case.
In Chapter 3, we studied directed graph packing problems for families of directed paths,
directed cycles a n d directed stars. T h e focus of the chapter was the statement and proof
of the main theorem of the thesis, which s ta tes t h a t the directed G-packing problem is
polynomial for t h e family G = {e, 6). To conclude, we provide a summary of the complexity of the problems presented in this
tliesis. Recall t h a t a polynomial problem can be solvecl by a polynomial-time algorithm.
\vhether such an algorithm has been found or not is another question! JVP-completeness is
a classification of problems of a certain degree of difficulty. Why is i t interesting t o know
t h a t a problem is one of these difficult types? Well, t o answer t h a t , we defer t o a nicely
stated explanation from ['24].
Indeed, discovering t h a t a problem is NP-comple te is usually just t h e beginning
of work on t h e problem. . . . However, t h e knowledge t h a t i t is JVP-complete does
provide valuable information a b o u t what lines of approach have t h e potential
of being most productive. Certainly the search for a n efficient, exact algorithm
should be accorded low priority. It is now more appropr ia te t o concentrate on
other, less ambitious, approaches. For example, you might look for efficient
algorithms t h a t solve various special cases of t h e general problem. You might
look for algorithms tha t , though not guaranteed t o run quickly. seem likely t o d o
s o most of the time. . . . In shor t , t he primary application of t h e theory of JVP- completeness is to assist algorithm designers in directing their problem-solving
efforts toward those approaches t h a t have t h e greatest likelihood of leading t o
usefuI algorithms.
Table 4.1: Complexity of the packing problem for families of cliques
Family
{Ir'l) { It-2 1 {a ' u p I&) {ICn}, n 2 3 { I(-2, K3 ) G {K1, &, K3, . . . }, €
G o r I<2 E G G {Ii-19f<2,K31...}9 K1, 1-2 4 G
Polynomial 0
0
0
N P - c o m p l e t e Reference
[20, 19, '221
[361 135, 1361
P I [ l i , 27, 29, 361
~ 9 1
CW.4 PTE R 4. CONCL USlON
Table 4.2: Complexity of the packing problem for families of complete bipart i te graphs
Table 4.3: Complexity of t h e packing problem for families of paths
Family , {sl, s2. - - - . & ) S = {S1,S2, . . . , I S not a sequential s e t of s t a r s
Table 4.4: Complexity of t h e packing problem for families of cycles
Polynomial
Family { P k ) , k > '2 I 9 7 P2)
N P - c o m p l e t e
Polynomial
Reference
128, 301 ['28, 301
[28, 30, 361
Family {C;, C4, . . . ) (Two-factors)
{C3?C5,C6? - - ) {ck, CS, . . . 1 { C ~ , C S ~ - - a )
All o the r G {C3, C4,. . . 1 {c3,c4,c6~ c 7 - - - ) {C3,C5tC7, - - - ) {CS, c"l C71- -) (c6: c79 c 8 1 - -)
AfP-Complete
Polynomial JVP-Complete
Reference '
[3, 281
Reference [20, 22, 29, 391
P93 1291 P93 P ~ I E l 8 ] B8] [I8] R8]
Table 4.S: Complexity of the packing problem for mixed families
I Familv I l K 2 . H I , H hypomatchable {K2, H ) , H perfectly rnatch-
I hypomatchable, perfectly
[ matchable
Polynomial a
Table 4.6: Complexity of the packing problem in the directed case
Family
{PI ) {A) , k 2 2
- {PI, ~ 2 )
{PI, q), for j odd, j > 3 {PI, Pi}, for j even, j 2 4 B c { A , . . ., PLL.. .}, 26-2 2 1 > k , k 2 2 7 f i + ~ @ G {ck), k > 3
4
{Sk), k L 2
Reference
[I21
Polynomial I JVP-Complete a
a
a
a
a
a
a
CW.4 PTER 4. CONCL USION
Much remains t o be studied in the area of directed graph packings. For example. packing
problems with families of bipartite or complete bipartite directed graphs, families of two or
more directed cycles or directed stars have not been considered. We hope that the work in
this thesis provides a helpful beginning t o the study of the directed graph packing problem.
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