Locally constrained graph homomorphisms

Jan KratochvílJan Kratochvíl

Charles University, PragueCharles University, Prague

Outline of the talk

Graph Graph hhomomorphismomomorphism Local constraints - Local constraints -

graph covers graph covers partial covers – frequency partial covers – frequency assignment role assignmentsassignment role assignments

Complexity results and questionsComplexity results and questions

1. Graph homomorphism

Edge preserving vertex mapping between Edge preserving vertex mapping between graphs graphs G G and and HH

f f : : VV((GG) ) VV((HH) s.t.) s.t.

uv uv EE((GG) ) ff((uu))ff((vv)) EE((HH) )

u

v

f(u)

f(v)f

G H

HH-COLORING-COLORINGInput: A graph Input: A graph GG..

Question: Question: homomorphism homomorphism GG H H??

Thm (Hell, NeThm (Hell, Nešetřilšetřil)): : HH-COLORING-COLORING is is polynomial for polynomial for HH bipartite and NP-complete bipartite and NP-complete otherwise.otherwise.

2. Local constraints

For every For every uu VV((GG),),

ff((NNGG((uu)) )) NNHH((ff((uu))))

u

f

f(u)

G H

Definition: A homomorphism Definition: A homomorphism ff : : G G HH

is calledis called

bijectivebijective

locallylocally injective injective if for every if for every u u VV((GG))

surjectivesurjective

the restricted mapping the restricted mapping f f :: N NGG((uu)) )) NNHH((ff((uu))))

bijectivebijective

isis injective injective ..

surjectivesurjective

2. Locally constrained homomorphisms loc. bijective = graph coversloc. bijective = graph covers loc. injective = partial covers = generalized loc. injective = partial covers = generalized

frequency assignmentfrequency assignment loc. surjective = role assignmentloc. surjective = role assignment

computational complexitycomputational complexity

2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of Topological graph theory, construction of

highly symmetric graphs (Biggs, Conway)highly symmetric graphs (Biggs, Conway) Degree preservingDegree preserving

2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of Topological graph theory, construction of

highly symmetric graphs (Biggs, Conway)highly symmetric graphs (Biggs, Conway) Degree preservingDegree preserving Local computation (Angluin, Courcelle)Local computation (Angluin, Courcelle)

2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of Topological graph theory, construction of

highly symmetric graphs (Biggs, Conway)highly symmetric graphs (Biggs, Conway) Degree preservingDegree preserving Local computation (Angluin, Courcelle)Local computation (Angluin, Courcelle) Degree partition preservingDegree partition preserving

Degree partition – the coarsest partition Degree partition – the coarsest partition

VV((GG) ) = V= V1 1 VV2 2 … … VVt t s.t.s.t.

there exist numbers there exist numbers rrij ij s.t.s.t.

| | NN((vv) ) VVj j | = | = rrij ij for every for every v v VVi i ..

2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of Topological graph theory, construction of

highly symmetric graphs (Biggs, Conway)highly symmetric graphs (Biggs, Conway) Degree preservingDegree preserving Local computation (Angluin, Courcelle)Local computation (Angluin, Courcelle) Degree partition preservingDegree partition preserving Finite planar coversFinite planar covers

Conjecture (Negami): A graph has a finite Conjecture (Negami): A graph has a finite planar cover if and only if it is projective planar cover if and only if it is projective planar.planar.

Conjecture (Negami): A graph has a finite Conjecture (Negami): A graph has a finite planar cover if and only if it is projective planar cover if and only if it is projective planar.planar.

Attempts to prove via forbidden minors for Attempts to prove via forbidden minors for projective planar graphs (Negami, Fellows, projective planar graphs (Negami, Fellows, Archdeacon, HliArchdeacon, Hliněnýněný))

Conjecture (Negami): A graph has a finite Conjecture (Negami): A graph has a finite planar cover if and only if it is projective planar cover if and only if it is projective planar.planar.

Attempts to prove via forbidden minors for Attempts to prove via forbidden minors for projective planar graphs (Negami, Fellows, projective planar graphs (Negami, Fellows, Archdeacon, HliArchdeacon, Hliněnýněný))

True if True if KK1,2,2,2 1,2,2,2 does not have a finite planar does not have a finite planar cover.cover.

2.2 Locally injective homomorphisms = partial covers

Observation: A graph Observation: A graph G G allows a locally allows a locally injective homomorphism into a graph injective homomorphism into a graph HH iff iff GG is an induced subgraph of a graph is an induced subgraph of a graph G’ G’ which covers which covers H H fully.fully.

2.2 Locally injective homomorphisms = generalized frequency assignment

L(2,1)-labelings of graphs

(Roberts; Griggs, Yeh;(Roberts; Griggs, Yeh;

Georges, Mauro; Sakai;Georges, Mauro; Sakai;

KrKráál, l, ŠkrekovskiŠkrekovski))

L(2,1)-labelings of graphs

f: Vf: V((GG)) {0,1,2{0,1,2,…,k,…,k}}

uv uv EE((GG) ) | |ff((uu) – ) – ff((vv)| )| 2 2

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

L(2,1)-labelings of graphs

f: Vf: V((GG)) {0,1,2{0,1,2,…,k,…,k}}

uv uv EE((GG) ) | |ff((uu) – ) – ff((vv)| )| 2 2

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

||ff((uu) – ) – ff((vv)| )| 1 1

L(2,1)-labelings of graphs

f: Vf: V((GG)) {0,1,2{0,1,2,…,k,…,k}}

uv uv EE((GG) ) | |ff((uu) – ) – ff((vv)| )| 2 2

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

||ff((uu) – ) – ff((vv)| )| 1 1

LL(2,1)(2,1)((GG)) = = min such min such kk

L(2,1)-labelings of graphs

NP-complete for every fixed NP-complete for every fixed k k 4 (Fiala, 4 (Fiala, Kloks, JK)Kloks, JK)

Polynomial for graphs of bounded tree-Polynomial for graphs of bounded tree-width (when width (when k k fixed)fixed)

L(2,1)-labelings of graphs

NP-complete for every fixed NP-complete for every fixed k k 4 (Fiala, Kloks, 4 (Fiala, Kloks, JK)JK)

Polynomial for graphs of bounded tree-width Polynomial for graphs of bounded tree-width (when (when k k fixed)fixed)

Polynomial for trees when Polynomial for trees when k k part of input (Chang, part of input (Chang, Kuo) Kuo)

Open for graphs of bounded tree-width (when Open for graphs of bounded tree-width (when k k part of input)part of input)

H(2,1)-labelings of graphs

(Fiala, JK 2001)(Fiala, JK 2001)

H(2,1)-labelings of graphs

(Fiala, JK 2001)(Fiala, JK 2001)

CCkk(2,1)-labelings have been considered by (2,1)-labelings have been considered by

Leese et al.Leese et al.

H(2,1)-labelings of graphs

f: Vf: V((GG)) V V((HH))

uv uv EE((GG) ) ddH H (( ff((uu), ), ff((vv)) )) 2 2

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

H(2,1)-labelings of graphs

f: Vf: V((GG)) V V((HH))

uv uv EE((GG) ) ddH H (( ff((uu), ), ff((vv)) )) 2 2

ff((uu))ff((vv) ) EE((HH) )

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

H(2,1)-labelings of graphs

f: Vf: V((GG)) V V((HH))

uv uv EE((GG) ) ff((uu))ff((vv) ) EE(-(-HH) )

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

H(2,1)-labelings of graphs

f: Vf: V((GG)) V V((HH))

uv uv EE((GG) ) ff((uu))ff((vv) ) EE(-(-HH) )

homomorphism from homomorphism from G G to -to -HH

ddGG((u,vu,v) = 2 ) = 2 ff((uu) ) ff((vv))

locally injective locally injective

H(2,1)-labelings of graphs

=

locally injective homomorphismsinto –H

L2,1(G) k

iff

G allows a Pk+1(2,1)-labeling

iff

G allows a locally injective homomorphism into -Pk+1 .

2.3 Locally surjective homomorphisms = role assignemts

Application in sociology – target vertices are Application in sociology – target vertices are roles in community, preimages are roles in community, preimages are members of a social groupmembers of a social group

3. Computational complexity

HH-COLORING-COLORINGInput: A graph Input: A graph GG..

Question: Question: homomorphism homomorphism GG H H??

Thm (Hell, NeThm (Hell, Nešetřilšetřil)): : HH-COLORING-COLORING is is polynomial for polynomial for HH bipartite and NP-complete bipartite and NP-complete otherwise.otherwise.

3.1 Locally surjective

HH--ROLE-ASSIGNMENTROLE-ASSIGNMENT

Input: A graph Input: A graph GG..

Question: Question: locally surjective locally surjective homomorphism homomorphism GG H H??

Thm (Kristiansen, Telle 2000; Thm (Kristiansen, Telle 2000; Fiala, Paulusma Fiala, Paulusma 20022002)): : HH--ROLE-ASSIGNMENT is polynomial ROLE-ASSIGNMENT is polynomial for for connected connected HH with at most 3 vertices and NP- with at most 3 vertices and NP-complete otherwise.complete otherwise.

3.2 Locally bijective

HH-COVER-COVERInput: A graph Input: A graph GG..

Question: Question: locally bijective homomorphism locally bijective homomorphism GG H H??

Complexity of H-COVER

Bodlaender 1989Bodlaender 1989 Abello, Fellows, Stilwell 1991Abello, Fellows, Stilwell 1991 JK, Proskurowski, Telle 1994, 1996, 1997JK, Proskurowski, Telle 1994, 1996, 1997 JiJiří ří FFialaiala 2000 2000

Complexity of H-COVER

NP-complete for NP-complete for kk-regular graphs -regular graphs HH ( (kk33))

Complexity of H-COVER

NP-complete for NP-complete for kk-regular graphs -regular graphs HH ( (kk33)) Polynomial for graphs with at most 2 Polynomial for graphs with at most 2

vertices in each block of the degree vertices in each block of the degree partitionpartition

Complexity of H-COVER

NP-complete for NP-complete for kk-regular graphs -regular graphs HH ( (kk33)) Polynomial for graphs with at most 2 Polynomial for graphs with at most 2

vertices in each block of the degree vertices in each block of the degree partitionpartition

Polynomial for graphs arising from affine Polynomial for graphs arising from affine mappingsmappings

Complexity of H-COVER NP-complete for NP-complete for kk-regular graphs -regular graphs HH ( (kk33)) Polynomial for graphs with at most 2 Polynomial for graphs with at most 2

vertices in each block of the degree partitionvertices in each block of the degree partition Polynomial for graphs arising from affine Polynomial for graphs arising from affine

mappingsmappings Polynomial for Theta graphs (based on Polynomial for Theta graphs (based on

KKönig-Hall theoremönig-Hall theorem))

Theorem (KPT): Theorem (KPT): GG covers covers ((aa11nn11,a,a22

nn22,…,a,…,akknnkk) if ) if

and only if and only if GG contains only vertices of contains only vertices of degrees 2 and degrees 2 and d = nd = n1 1 + n+ n22 + … + n + … + nk, k, and the and the

vertices of degree vertices of degree dd can be colored by two can be colored by two colors colors red red and and blueblue so that each one is so that each one is connected by exactly connected by exactly nni i paths of length paths of length aai i to to

the vertices of the opposite color.the vertices of the opposite color.

G

((aa11nn11,a,a22

nn22,…,a,…,akknnkk))

((aaiinnii))

G’

Complexity of H-COVER NP-complete for NP-complete for kk-regular graphs -regular graphs HH ( (kk33)) Polynomial for graphs with at most 2 Polynomial for graphs with at most 2

vertices in each block of the degree partitionvertices in each block of the degree partition Polynomial for graphs arising from affine Polynomial for graphs arising from affine

mappingsmappings Polynomial for Theta graphs (based on Polynomial for Theta graphs (based on

KKönig-Hall theoremönig-Hall theorem)) Full characterization for Weight graphsFull characterization for Weight graphs

WW((aa11nn11,a,a22

nn22,…,a,…,akknnkk;a;a11

ll11,a,a22ll22,…,a,…,akk

llk k ;a;a11mm11,a,a22

mm22,…,a,…,akkmmkk))

Theorem (KPT): The W-COVER problem is Theorem (KPT): The W-COVER problem is NP-complete if NP-complete if

nni i = m= mi i for all for all i, i, andand

nni i . l. li i > 0 for some > 0 for some ii

and polynomial time solvable otherwise.and polynomial time solvable otherwise.

3.3 Locally injective

HH-PARTIAL-COVER-PARTIAL-COVERInput: A graph Input: A graph GG..

Question: Question: locally injective homomorphism locally injective homomorphism GG H H??

TheoremTheorem (FK): If (FK): If GG and and HH have the same have the same degree refinement matrix, then every degree refinement matrix, then every locally injective homomorphism locally injective homomorphism ff : : G G H H is locally bijective. is locally bijective.

TheoremTheorem (FK): If (FK): If GG and and HH have the same have the same degree refinement matrix, then every degree refinement matrix, then every locally injective homomorphism locally injective homomorphism ff : : G G H H is locally bijective. is locally bijective.

Corollary: Corollary: HH-COVER -COVER HH-PARTIAL--PARTIAL-COVERCOVER

TheoremTheorem (FK): If (FK): If GG and and HH have the same have the same

degree refinement matrix, then every locally degree refinement matrix, then every locally injective homomorphism injective homomorphism ff : : G G H H is is locally bijective. locally bijective.

Corollary: Corollary: HH-COVER -COVER HH-PARTIAL--PARTIAL-COVERCOVER

Corollary: Corollary: CCkk(2,1)-labeling is NP-complete for (2,1)-labeling is NP-complete for

every every k k 6. 6.

Partial covers of Theta graphs

Partial covers of Theta graphs

ThmThm (Fiala, JK) (Fiala, JK): : ((aakk,b,bmm)-PARTIAL-COVER is)-PARTIAL-COVER is

- polynomial if - polynomial if a,b a,b are oddare odd

- NP-complete if - NP-complete if a-b a-b is oddis odd

Partial covers of Theta graphs

ThmThm (FK) (FK): : ((aakk,b,bmm)-PARTIAL-COVER is)-PARTIAL-COVER is

- polynomial if - polynomial if a,b a,b are oddare odd

- NP-complete if - NP-complete if a-b a-b is oddis odd

Thm (Fiala, JK, PThm (Fiala, JK, Pórór): ): ((a,b,ca,b,c)-PARTIAL-COVER is)-PARTIAL-COVER is

NP-complete if NP-complete if a,b,c a,b,c are distinct odd integersare distinct odd integers

Partial covers of Theta graphs

Thm: Thm: ((aakk,b,bmm)-PARTIAL-COVER is)-PARTIAL-COVER is

- polynomial if - polynomial if a,b a,b are oddare odd

- NP-complete if - NP-complete if a-b a-b is oddis odd

Thm: Thm: ((a,b,ca,b,c)-PARTIAL-COVER is)-PARTIAL-COVER is

NP-complete if NP-complete if a,b,c a,b,c are distinct odd integersare distinct odd integers

ThmThm (FK) (FK): : ((a,b,ca,b,c)-PARTIAL-COVER is)-PARTIAL-COVER is

NP-complete if NP-complete if a+b|ca+b|c

Proof

Given cubic bipartite graph Given cubic bipartite graph GG, it is NP-complete to , it is NP-complete to decide if the vertices of decide if the vertices of GG can be bicolored so that can be bicolored so that every vertex has exactly one neighbor of the other every vertex has exactly one neighbor of the other color (color (WW(1;1;1)-COVER)(1;1;1)-COVER)..

Proof

Given cubic bipartite graph Given cubic bipartite graph GG, it is NP-complete to , it is NP-complete to decide if the vertices of decide if the vertices of GG can be bicolored so that can be bicolored so that every vertex has exactly one neighbor of the other every vertex has exactly one neighbor of the other color (color (WW(1;1;1)-COVER)(1;1;1)-COVER)..

Given Given GG, construct , construct G’G’ by replacing its edges by paths by replacing its edges by paths of length of length c. c.

c = a + b + a + b + … + a + b c = a + b + a + b + … + a + b

= b + a + b + a + … + b + a= b + a + b + a + … + b + a

= c = c

G’

G’

G

G

G’

ProofGiven cubic bipartite graph Given cubic bipartite graph GG, it is NP-complete to decide if , it is NP-complete to decide if

the vertices of the vertices of GG can be bicolored so that every vertex has can be bicolored so that every vertex has exactly one neighbor of the other color (exactly one neighbor of the other color (WW(1;1;1)-COVER)(1;1;1)-COVER)..

Construct Construct G’G’ by replacing its edges by paths of length by replacing its edges by paths of length c. c.

Then Then G’G’ partially covers partially covers ((a,b,ca,b,c) iff ) iff GG covers covers WW(1;1;1).(1;1;1).

(1,2,3)

-(1,2,3) = P5

Eq:

(1,2,3)-PARTIAL-COVER (1,2,3)-PARTIAL-COVER PP55(2,1)-labeling(2,1)-labeling

LL(2,1)(2,1)((GG) ) 4 4

Eq:

(1,2,3)-PARTIAL-COVER (1,2,3)-PARTIAL-COVER PP55(2,1)-labeling(2,1)-labeling

LL(2,1)(2,1)((GG) ) 4 4

And hence all NP-complete.And hence all NP-complete.

Questions – Partial cover

More than 3 paths - More than 3 paths - ((a,b,c,d,…a,b,c,d,…)) Multiple lengths - Multiple lengths - ((aann,b,bmm,c,ckk))

Questions – Partial cover

More than 3 paths - More than 3 paths - ((a,b,c,d,…a,b,c,d,…)) Multiple lengths - Multiple lengths - ((aann,b,bmm,c,ckk))

Beyond Theta graphs – Beyond Theta graphs –

HH-PARTIAL-COVER is conjectured-PARTIAL-COVER is conjectured

NP-complete for NP-complete for HH containing a containing a

subdivision of subdivision of KK44

Questions – Partial cover

Dichotomy ?Dichotomy ? Plausible conjecture ?Plausible conjecture ?

Questions – Cover

Dichotomy ?Dichotomy ?

Questions – Cover

Dichotomy ?Dichotomy ? Perhaps affine graphs and graphs with Perhaps affine graphs and graphs with

Unique Neighbor Property are the only Unique Neighbor Property are the only polynomial cases for polynomial cases for HH-COVER-COVER

Questions – Cover and Partial Cover

Planar instancesPlanar instances

Thank youThank you

6th Czech-Slovak International Symposium

on Graphs and Combinatorics

PraguePrague,, July 10-15, 2006 July 10-15, 2006

In honor of 60In honor of 60thth birthday of Jarik Ne birthday of Jarik Nešetřilšetřil