Locally constrained graph homomorphisms

1
Locally constrained graph homomorphisms

• Jan Kratochvíl
• Charles University, Prague

2
Outline of the talk

• Graph homomorphism
• Local constraints -
  graph covers
  partial
  covers frequency assignment role
  assignments
• Complexity results and questions

3
1. Graph homomorphism

• Edge preserving vertex mapping between graphs G
  and H
• f V(G) ? V(H) s.t.
• uv ? E(G) ? f(u)f(v) ? E(H)

4
v
f(v)
f
u
f(u)
H
G
5
(No Transcript)
6

• H-COLORING
• Input A graph G.
• Question ? homomorphism G? H?
• Thm (Hell, Neetril) H-COLORING is polynomial
  for H bipartite and NP-complete otherwise.

7
2. Local constraints

• For every u ? V(G),
• f(NG(u)) ? NH(f(u))

8
f
f(u)
u
H
G
9

• Definition A homomorphism f G ? H
• is called
• bijective
• locally injective if for every u ? V(G)
• surjective
• the restricted mapping f NG(u)) ? NH(f(u))
• bijective
• is injective .
• surjective

10
2. Locally constrained homomorphisms

• loc. bijective graph covers
• loc. injective partial covers generalized
  frequency assignment
• loc. surjective role assignment
• computational complexity

11
2.1 Locally bijective homomorphisms graph covers

• Topological graph theory, construction of highly
  symmetric graphs (Biggs, Conway)
• Degree preserving

12
(No Transcript)
13
2.1 Locally bijective homomorphisms graph covers

• Topological graph theory, construction of highly
  symmetric graphs (Biggs, Conway)
• Degree preserving
• Local computation (Angluin, Courcelle)

14
2.1 Locally bijective homomorphisms graph covers

• Topological graph theory, construction of highly
  symmetric graphs (Biggs, Conway)
• Degree preserving
• Local computation (Angluin, Courcelle)
• Degree partition preserving

15

• Degree partition the coarsest partition
• V(G) V1 ? V2 ? ? Vt s.t.
• there exist numbers rij s.t.
• N(v) ? Vj rij for every v ? Vi .

16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
2.1 Locally bijective homomorphisms graph covers

• Topological graph theory, construction of highly
  symmetric graphs (Biggs, Conway)
• Degree preserving
• Local computation (Angluin, Courcelle)
• Degree partition preserving
• Finite planar covers

26
(No Transcript)
27
(No Transcript)
28

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

29

• Conjecture (Negami) A graph has a finite planar
  cover if and only if it is projective planar.
• Attempts to prove via forbidden minors for
  projective planar graphs (Negami, Fellows,
  Archdeacon, Hlinený)

30

• Conjecture (Negami) A graph has a finite planar
  cover if and only if it is projective planar.
• Attempts to prove via forbidden minors for
  projective planar graphs (Negami, Fellows,
  Archdeacon, Hlinený)
• True if K1,2,2,2 does not have a finite planar
  cover.

31
2.2 Locally injective homomorphisms partial
covers

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

32
2.2 Locally injective homomorphisms generalized
frequency assignment
33
L(2,1)-labelings of graphs

• (Roberts Griggs, Yeh
• Georges, Mauro Sakai
• Král, krekovski)

34
L(2,1)-labelings of graphs

• f V(G) ? 0,1,2,,k
• uv ? E(G) ? f(u) f(v) ? 2
• dG(u,v) 2 ? f(u) ? f(v)

35
L(2,1)-labelings of graphs

• f V(G) ? 0,1,2,,k
• uv ? E(G) ? f(u) f(v) ? 2
• dG(u,v) 2 ? f(u) ? f(v)
• f(u) f(v) ? 1

36
L(2,1)-labelings of graphs

• f V(G) ? 0,1,2,,k
• uv ? E(G) ? f(u) f(v) ? 2
• dG(u,v) 2 ? f(u) ? f(v)
• f(u) f(v) ? 1
• L(2,1)(G) min such k

37
(No Transcript)
38
(No Transcript)
39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
(No Transcript)
43
L(2,1)-labelings of graphs

• NP-complete for every fixed k ? 4 (Fiala, Kloks,
  JK)
• Polynomial for graphs of bounded tree-width (when
  k fixed)

44
L(2,1)-labelings of graphs

• NP-complete for every fixed k ? 4 (Fiala, Kloks,
  JK)
• Polynomial for graphs of bounded tree-width (when
  k fixed)
• Polynomial for trees when k part of input (Chang,
  Kuo)
• Open for graphs of bounded tree-width (when k
  part of input)

45
H(2,1)-labelings of graphs

• (Fiala, JK 2001)

46
H(2,1)-labelings of graphs

• (Fiala, JK 2001)
• Ck(2,1)-labelings have been considered by Leese
  et al.

47
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? dH ( f(u), f(v)) ? 2
• dG(u,v) 2 ? f(u) ? f(v)

48
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? dH ( f(u), f(v)) ? 2
• ? f(u)f(v) ? E(H)
• dG(u,v) 2 ? f(u) ? f(v)

49
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? f(u)f(v) ? E(-H)
• dG(u,v) 2 ? f(u) ? f(v)

50
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? f(u)f(v) ? E(-H)
• homomorphism from G to
  -H
• dG(u,v) 2 ? f(u) ? f(v)
• locally injective

51
H(2,1)-labelings of graphs
locally injective homomorphismsinto H
52
L2,1(G) ? k iff G allows a
Pk1(2,1)-labeling iff G allows a locally
injective homomorphism into -Pk1 .
53
2.3 Locally surjective homomorphisms role
assignemts

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

54
3. Computational complexity

• H-COLORING
• Input A graph G.
• Question ? homomorphism G? H?
• Thm (Hell, Neetril) H-COLORING is polynomial
  for H bipartite and NP-complete otherwise.

55
3.1 Locally surjective

• H-ROLE-ASSIGNMENT
• Input A graph G.
• Question ? locally surjective homomorphism
  G? H?
• Thm (Kristiansen, Telle 2000 Fiala, Paulusma
  2002) H-ROLE-ASSIGNMENT is polynomial for
  connected H with at most 3 vertices and
  NP-complete otherwise.

56
3.2 Locally bijective

• H-COVER
• Input A graph G.
• Question ? locally bijective homomorphism G?
  H?

57
Complexity of H-COVER

• Bodlaender 1989
• Abello, Fellows, Stilwell 1991
• JK, Proskurowski, Telle 1994, 1996, 1997
• Jirí Fiala 2000

58
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)

59
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition

60
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition
• Polynomial for graphs arising from affine
  mappings

61
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition
• Polynomial for graphs arising from affine
  mappings
• Polynomial for Theta graphs (based on König-Hall
  theorem)

62
(No Transcript)
63
(No Transcript)
64

• Theorem (KPT) G covers ?(a1n1,a2n2,,aknk) if
  and only if G contains only vertices of degrees 2
  and d n1 n2 nk, and the vertices of
  degree d can be colored by two colors red and
  blue so that each one is connected by exactly ni
  paths of length ai to the vertices of the
  opposite color.

65
G
?(a1n1,a2n2,,aknk)
66
(No Transcript)
67
?(aini)
G
68
(No Transcript)
69
(No Transcript)
70
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition
• Polynomial for graphs arising from affine
  mappings
• Polynomial for Theta graphs (based on König-Hall
  theorem)
• Full characterization for Weight graphs

71
W(a1n1,a2n2,,aknka1l1,a2l2,,aklk
a1m1,a2m2,,akmk)
72

• Theorem (KPT) The W-COVER problem is NP-complete
  if
• ni mi for all i, and
• ni . li gt 0 for some i
• and polynomial time solvable otherwise.

73
3.3 Locally injective

• H-PARTIAL-COVER
• Input A graph G.
• Question ? locally injective homomorphism
  G? H?

74

• Theorem (FK) If G and H have the same degree
  refinement matrix, then every locally injective
  homomorphism f G ? H is locally
  bijective.

75

• Theorem (FK) If G and H have the same degree
  refinement matrix, then every locally injective
  homomorphism f G ? H is locally
  bijective.
• Corollary H-COVER ? H-PARTIAL-COVER

76

• Theorem (FK) If G and H have the same degree
  refinement matrix, then every locally injective
  homomorphism f G ? H is locally bijective.
• Corollary H-COVER ? H-PARTIAL-COVER
• Corollary Ck(2,1)-labeling is NP-complete for
  every k ? 6.

77
Partial covers of Theta graphs
78
Partial covers of Theta graphs

• Thm (Fiala, JK) ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd

79
Partial covers of Theta graphs

• Thm (FK) ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd
• Thm (Fiala, JK, Pór) ?(a,b,c)-PARTIAL-COVER is
• NP-complete if a,b,c are distinct odd
  integers

80
Partial covers of Theta graphs

• Thm ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd
• Thm ?(a,b,c)-PARTIAL-COVER is
• NP-complete if a,b,c are distinct odd
  integers
• Thm (FK) ?(a,b,c)-PARTIAL-COVER is
• NP-complete if abc

81
Proof

• Given cubic bipartite graph G, it is NP-complete
  to decide if the vertices of G can be bicolored
  so that every vertex has exactly one neighbor of
  the other color (W(111)-COVER).

82
Proof

• Given cubic bipartite graph G, it is NP-complete
  to decide if the vertices of G can be bicolored
  so that every vertex has exactly one neighbor of
  the other color (W(111)-COVER).
• Given G, construct G by replacing its edges by
  paths of length c.

83
(No Transcript)
84
(No Transcript)
85

• c a b a b a b
• b a b a b a
• c

86
G
87
G
G
88
G
89
G
90
Proof

• Given cubic bipartite graph G, it is NP-complete
  to decide if the vertices of G can be bicolored
  so that every vertex has exactly one neighbor of
  the other color (W(111)-COVER).
• Construct G by replacing its edges by paths of
  length c.
• Then G partially covers ?(a,b,c) iff
• G covers W(111).

91
?(1,2,3)
92
(No Transcript)
93
-?(1,2,3) P5
94
Eq

• ?(1,2,3)-PARTIAL-COVER
• P5(2,1)-labeling
• L(2,1)(G) ? 4

95
Eq

• ?(1,2,3)-PARTIAL-COVER
• P5(2,1)-labeling
• L(2,1)(G) ? 4
• And hence all NP-complete.

96
Questions Partial cover

• More than 3 paths - ?(a,b,c,d,)
• Multiple lengths - ?(an,bm,ck)

97
Questions Partial cover

• More than 3 paths - ?(a,b,c,d,)
• Multiple lengths - ?(an,bm,ck)
• Beyond Theta graphs
• H-PARTIAL-COVER is conjectured
• NP-complete for H containing a
• subdivision of K4

98
Questions Partial cover

• Dichotomy ?
• Plausible conjecture ?

99
Questions Cover

• Dichotomy ?

100
Questions Cover

• Dichotomy ?
• Perhaps affine graphs and graphs with Unique
  Neighbor Property are the only polynomial cases
  for H-COVER

101
Questions Cover and Partial Cover

• Planar instances

102

• Thank you

103
6th Czech-Slovak International Symposiumon
Graphs and Combinatorics

• Prague, July 10-15, 2006
• In honor of 60th birthday of Jarik Neetril