Twenty Years of EPT Graphs: From Haifa to Rostock

Martin Charles Golumbic

Caesarea Rothschild Institute

University of Haifa

With thanks to my research collaborators :

Robert Jamison, Marina Lipshteyn, Michal Stern

Bell Labs in New Jersey (Spring 1981)

John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls?

The Story Begins

Bell Labs in New Jersey (Spring 1981)

John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls?

The Story Begins

Bell Labs in New Jersey (Spring 1981)

John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls?

The Story Begins

•A call is a path between a pair of nodes.

• A typical example of a type of intersection graph.

• Intersection here means “share an edge”.

•Coloring this intersection graph is scheduling the calls.

An Olive Tree Network

Vertex Intersection Graphs of Paths in a Tree (VPT) Edge Intersection Graphs of Paths in a Tree (EPT)

Each vertex v in V(GVPT) and V(GEPT) corresponds to a path Pv in T.

(x,y) EVPT paths Px and Py intersect on at least one vertex in T.

(x,y) EEPT paths Px and Py intersect on at least one edge in T.

Representation P of paths in a tree T.

GVPT(P)

Pb

Pa

Pd

Pb

a

bd

cPc

GEPT(P)

a

d b

c

For VPT-representation Pa and Pb intersect. For EPT-representation: Pa and Pb do not intersect.

For both VPT-representation and EPT-representation Pa and Pb intersect.

a a

bb 1 vertex

a a

bb 1 edge

Vertex and Edge Intersections of Paths

Theorem. Chordal graphs vertex intersection graphs of

subtrees of a tree. [Buneman], [Gavril], [Walter]

A graph G is chordal if every cycle of size 4 has a chord,

i.e., G has no induced chordless cycles Cm for m 4 .

Grepresentation of G on tree T

Chordal Graphs

VPT Graphs are Chordal EPT Graphs are Not Chordal

• A path is a subtree, thereforeVPT graphs (i.e., path graphs) are chordal.

• However,

EPT graphs may have chordless cycles of any size.

A First Observation (Cycles)

An EPT representation of C6

called a “6-pie”. 6

3

2

1

4

5

Chordless cycles have a unique EPT representation.

A First Observation (Cycles)

An EPT representation of C6

called a “6-pie”. 6

3

2

1

4

5Theorem (Golumbic Jamison 1985):

Let P be an EPT representation of G.

If G contains a chordless cycle Cm (m 4), then P contains an m-pie

representing the cycle.

Chordless cycles have a unique EPT representation.

Restricting the degree of the host tree

Remark. If m is the maximum degree in T, then the EPT graph has no chordless (m+1) cycles (or larger).

Restricting the degree of the host tree

Remark. If m is the maximum degree in T, then the EPT graph has no chordless (m+1) cycles (or larger).

Corollary:

If P is an EPT representation of G on a degree 3 tree T. Then G is chordal graph.

Restricting the degree of the host tree

C6

aa

a

b

b

bc

c

cd

d

d

Example. The graph C6 requires degree 5.

A 4-pie on a,b,c,d

Restricting the degree of the host tree

C6

aa

a

b

b

bc

c

cd

d

d

x

xx

y

y

y

Example. The graph C6 requires degree 5.

Now add x and y

A Second Observation (Cliques)

Two EPT representations of K6

called a “claw clique” or “edge clique”. 6

3

2

1

4

5

All share a common edgeedge

All share some edge of the clawclaw

A Second Observation (Cliques)

Two EPT representations of K6

called a “claw clique” or “edge clique”. 6

3

2

1

4

5Theorem (Golumbic Jamison 1985):

Let P be an EPT representation of G.

If G contains a clique Km (m 3), then P contains either a claw or edge for it.

Cliques have exactly two possible EPT representations.

No Kissing

The No Kissing Lemma: If P is an EPT representation of G on a tree T, and u is any node of T. We may assume without loss of generality, and without increasing the degree of the tree, that all paths touching u continue through u.

No stopping. No kissing u.

No Kissing

The No Kissing Lemma: If P is an EPT representation of G on a tree T, and u is any node of T. We may assume without loss of generality, and without increasing the degree of the tree, that all paths touching u continue through u.

No stopping. No kissing u.

e

c

b

a

d

e

c

b

a

d

Create dummy

nodes and shorten a and e

Degree 3 host trees

If P is a deg3 EPT representation of G on a tree T, then applying the No Kissing Lemma construction to all nodes of degree 3 yields a deg3 VPT representation of G.

Degree 3 host trees

If P is a deg3 EPT representation of G on a tree T, then applying the No Kissing Lemma construction to all nodes of degree 3 yields a deg3 VPT representation of G.

i.e., deg3 EPT deg3 VPT deg3 EPT chordal EPT

Now let’s prove: chordal EPT deg3 EPT

Degree 3 host trees (continued)

Let P be any EPT representation of G on a tree T, and u any node of T of maximum degree d > 3.

1. Assume the no kissing lemma, and let U denote all paths passing through u.

2. Let x be a simplicial vertex of the induced subgraph GU . Thus, all paths in U share an edge with Px .

3. Perform the transformation:

4. Repeat for all nodes until max degree is 3.

x

y

zx

y

z

Degree 3 host trees (continued)

Theorem (1985): All four classes are equivalent:

chordal EPT deg3 EPT

VPT EPT deg3 VPT

What about degree 4?

Degree 3 host trees (continued)

Theorem (1985): All four classes are equivalent:

chordal EPT deg3 EPT

VPT EPT deg3 VPT

Theorem (2005, Golumbic, Lipshteyn, Stern):

weakly chordal EPT deg4 EPT

Degree 4 host trees

Definition Weakly Chordal Graph No induced Cm for m 5, and

no induced Cm for m 5.

Theorem [Hayward, Hoàng, Maffray 1989] G is weakly chordal if and only if every induced subgraph of G is either a clique or has a two-pair.

Weakly Chordal Graphs

A two-pair is a pair of vertices, such that every chordless path between them has length two edges.

Remark. If {x,y} is a two-pair, then the common

neighborhood of x and y is an (x,y)-minimal separator.

x y

{x,y} is a two-pair

yx

{x,y} is not a two-pair

Theorem [GLS 2005] A graph G has an EPT representation on a degree 4 tree if and only if G is a weakly chordal EPT graph.

Degree 4 Trees

Sketch of the proof.

() By the Pie Theorem, G has no induced Cm (m 5) nor C5 (=C5).

By our earlier example, C6 requires degree 5.

By a theorem of Golumbic and Jamison 1985, Cm (m 7) is not an EPT graph.

Degree 4 Trees (continued)

() Let P be an EPT representation of G on tree T with maximal degree d > 4, and let u be a node of degree d. We transform P into an EPT representation P´ on T´ with fewer vertices of degree d. The full proof follows by induction.

• Assume the no kissing lemma at u, and let U denote all paths passing through u.

• The induced subgraph GU is weakly chordal, so there are two cases:

GU is a clique or GU has a two-pair.

Degree 4 Trees (continued)

Case 1. GU is a clique. If it were a claw clique, then u would have degree 3, and we are done. Otherwise, GU is an edge clique, and all paths in U share an edge, say (v1,u). Perform the transformation:

u uwdummy

v1 v1

v2 v2

Degree 4 Trees (continued)

Case 2. GU has a two pair {x,y}. The common neighborhood S of {x,y} is a minimal separator and splits GU into (at least) 2 connected components: GX containing x and GY containing y.

The star edges centered at u, are now painted.

• The two contained in Px are red; those in Py blue.

• Propagate the coloring to other star edges via the paths Pz (z U \ S):

if Pz has one red edge, then paint its other star edge red;

if Pz has one blue edge, then paint its other star edge blue.

No star edges gets two colors!

Degree 4 Trees (continued)

Subcase 2a. GS is a clique.

Perform the transformation:

u'x

vxv

ys1

s1

s2

s2 yu''ux y

vxv

ys1

s1

s2

s2 u'x

vxv

ys1

s1

s2

s2 yu''

x yv

S

Degree 4 Trees (continued)

Subcase 2b. GS is not a clique.

There exists a path Pv (v U \ S) that contains only one of the edges (v1,u),(v2,u), (v3,u),(v4,u), say (v1,u). Let be the non-empty collection of such paths, which thus form an edge clique containing (v1,u).

Color the star edges as follows:

x yvS

uxv

xv

y

s

s2

s1

s1 v1

v2

v3

v4

2

y

Subcase 2b, continued. 1) .v1,u (is not colored.

2) .vi,u ,(i 5 is colored pink if it is contained in a path in .

3) .vi,u (is colored if is contained in a path that already has a pink edge.Lemma: The edges of Py are not colored.

u

y

s

s2

v2

v3

v4

2

y

x

s1

xs1

v112

w

v51 v6

3

v7

2

3

ux

y

s

s2

s1

v1

v2

v3

v4

2

y

v5

v6

v7

x

1

1

s1

2

2

3

3

Example.

[v1,u,v5] P1 [v1,u,v7] P2 [v5,u,v6] P3

= {P1,P2}

Q.E.D.

From Haifa to Rostock (Spring 1985?)

The Story Continues

Algorithmic Aspects of EPT Graphs

• The recognition and coloring problems of an EPT graph are NP-complete (Golumbic and Jamison, 1985).

• There is a 3/2-approximation algorithm for coloring EPT graphs. (Tarjan, 1985)

• On deg3EPT or deg4EPT graphs, coloring is polynomial since, respectively, they are chordal (GJ 1985) or weakly chordal (GLS 2005).

• Max-clique and Max-stable set are polynomial (GJ 1985).

[h,s,t] Graphs and Representations

A collection of subtrees of a tree T satisfying: h: T has maximum vertex degree h s: Each subtree has maximum vertex degree s

t: An edge (x,y) in G if Tx and Ty share t vertices

Interval graphs [2,2,1] EPT [, 2, 2]

Chordal graphs [,,1] [3,3,1] [3,3,2] (MS,JM)

[h,s,t] Graphs and Representations

A collection of subtrees of a tree T satisfying: h: T has maximum vertex degree h s: Each subtree has maximum vertex degree s

t: An edge (x,y) in G if Tx and Ty share t vertices

Interval graphs [2,2,1] EPT [, 2, 2]

Chordal graphs [,,1] [3,3,1] [3,3,2] (MS,JM)Any graph [,,2]

[,2,2] chordal [3,2,2] [3,2,1] (GJ)

[,2,2] weakly chordal [4,2,2] (GLS)

[h,s,t] Graphs and Representations

A collection of subtrees of a tree T satisfying: h: T has maximum vertex degree h s: Each subtree has maximum vertex degree s

t: An edge (x,y) in G if Tx and Ty share t vertices

Two Directions to Generalize EPT

1. Keep t = 2 (edge intersection) Increase s (generalizing to subtrees) but Bound max degree h.

2. Increase t (constant tolerence) Keep s = 2 (paths)

A New Characterization TheoremGolumbic, Lipshteyn, Stern [WG2006]:

The class [4,4,2]-graphs is equivalent to weakly chordal (K2,3, P6, 4P2, P2 P4 , H1, H2 , H3)-free.

H 2H

3H

1

e

a

b d

c

h

K2

K1

e

a

b d

c

h

K2K1

e

a

b d

c

h

K2

K1

K1 and K2 are cliques of size at most 2.

v

v

6P 2P 4P 24 P

v v v vv

v v v v

v v

2 ,3K

v v

v v

v v

v v

Forbidden Subgraphs

Definition of k-EPT Graphs The k-Edge Intersection Graphs of Paths in a Tree

(x,y) E paths Px and Py intersect on at least k edges in T.

Def. G is a k-EPT graph if G has a k-EPT representation.

k-EPT representation tree T of G G = (V,E) for k = 4 edges

a

bd

cPaPc Pd

Pb

k edges

k-EPT [, 2, k+1] (i.e., share k+1 vertices)k-EPT [, 2, k+1] (i.e., share k+1 vertices)

Examples of Intersections

For VPT representation: paths a and b intersect. For k-EPT representation, k>0: paths a and b do not intersect.

For VPT and 1-EPT representation: paths a and b intersect. For k-EPT representation, k>1: paths a and b do not intersect.

For VPT and k-EPT representation, k 4: paths a and b intersect. For k-EPT representation, k>4: paths a and b do not intersect.

a a

bb 1 vertex

a a

bb 1 edge

a a

bb k edges

Properties of k-EPT• 1-EPT k-EPT, for any fixed k > 1.

- Divide each edge into k edges, by adding k-1 dummy vertices.

•When restricted to degree 3 trees, the containment is also strict.

• 1-EPT k-EPT, for any fixed k > 1.

a

c

f b

de

ggP

Pa

Pd

P f

Pe Pc

Pb

k ed

ges

k edg

es

k edgesk edges

k-1 edges

New Properties of k-EPT

• VPT graphs are incomparable with k-EPT graphs, for any fixed k 1.

• When restricted to degree 3 trees,

VPT k-EPT, for any fixed k 2.

• Chordless cycles are degree 3 k-EPT for k 2.

Recognition of k-EPTImportant Properties

• Any maximal clique of a k-EPT graph is eithera k-edge clique or a k-claw clique.

• A k-EPT graph G has at most maximal cliques. k-edge clique k-claw clique

k edgesk edges k edges

k ed

ges

( )O |V(G)|3

Recognition of k-EPTBranch Graphs

Definition: Let C be a subset of vertices of G. The branch graph B(G/C):

G B(G/C)

Theorem: Let C be a maximal clique of a k-EPT graph G. Then the branch graph B(G/C) can be 3-colored.

C x

y

x

y

Recognition of k-EPTNP-Completeness

Theorem:It is an NP-complete problem to decide whether a VPT graph is a k-EPT graph.

Proof:

An arbitrary undirected graph H is 3-colorable iffa certain graph G=(V,E) is a k-EPT graph.

rqa qj

qiqb qc

pa pj

pipcpb

Qa Qj

QcQb Qi

Pij

Pac

a b

ci j

H

T

Pij path in T (i,j) E(H)

Qi edge in T i V(H)

{Pij }corresponds to a maximal clique C of the VPT graph G.

B(G/C) is isomorphic to H.

If G is VPT and k-EPT H is 3-colorable

If H is 3-colorable and G is VPT G is VPT and 1-EPT

G is k-EPT. Therefore,G is VPT and k-EPT G is VPT and H is 3-colorable

rqa qj

qiqb qc

pa pj

pipcpb

Qa Qj

QcQb Qi

Pij

Pac

T

a b

ci j

HPij path in T (i,j) E(H)

Qi edge in T i V(H)

Recognition of k-EPTCorollaries

• Corollary:

Recognizing whether an arbitrary graph is a k-EPT graph is an NP-complete problem.

• Corollary:

Let G be a VPT graph. Then

G is a 1-EPT graph iff G is a k-EPT graph,

(hence: iff G is chordal).

Coloring of k-EPT

• Theorem:

The problem of finding a minimum coloring of a k-EPT graph is NP-complete.

Same proof as Golumbic & Jamison (1985)

for the case k = 1.

Forbidden Subgraph

• Theorem:

The following graph is not a k-EPT graph, for any fixed k > 1.

Open Problem

• The relationships between k-EPT graphs and (k+1)-EPT graphs, for any fixed k.

• Is k-EPT (k+1)-EPT?

• We have graphs that are not 1-EPT

but are a (k+1)-EPT graph, for any fixed k.

C4 is not VPT but is k-EPT k1

D is k-EPT k 2 is not 1-EPT

Graph D

Orthodox Representations

A representation <PP,T> for G is orthodox if

• For each path, its endpoints are leaves

(leaf generated), and

• Two paths Pi, Pj share a leaf if and only if vertices i and j are adjacent in G.

Subtrees of a Tree

(i) (ii) (iii) McMorris & Scheinerman 1991

(iv) (v) Jamison & Mulder 2000

(,,1)

Orthodox Representations

orth(,2,1) orth(3,2,1) orth(3,2,2)

Theorem 6.8

The Complete Heirarchy

The Complete Heirarchy

6.4

6.7

6.8

The Complete Heirarchy

EPT*

k-EPT

1-EPT

k-EPT

weakly chordal

orthodoxEPT*

orthodoxk-EPT

orthodox1-EPT

1-EPTdegree 4

1-EPT

weakly chordal

orthodox 1-EPTdegree 3

orthodox VPTdegree 3

orthodoxVPT

1-EPTdegree 3

1-EPT

chordal

1-EPTVPTVPT

degree 3

EPT*

chordal

weaklychordal

F*, H, A

Fk

D, F1

T1

C4,C5

Pn, n>0

C4,C5

D, C5

C5

C5

F*

F*

D, C5

H, C4

A

T1

Fk

FkC4

C4

F1C4

F1

EPT*

weakly chordal

FkF*

y1 y2

x2

x1

x3

x4

x5

H

chordal

orthodox[3,3,1]

orthodox[3,3,2]

k-EPT

chordal

[3,3,1]

[3,3,2]

VPT

F*, F1, Fk

a

c

f b

de

g

DT1

a

cd

b2

3

4

1

F1

a1

2

3

4

1

a2 ak...

b1 b2 ... bk

c1 c2 ck...... dkd1 d2

1

23

45

a

b

c

d

Further Research

Characterize families of [h,s,t] graphs for various values of h, s and t.

Find intersection models as (h,s,t)-representations for known families of graphs.

weakly chordal [?, ?, ?]

Thank You!