1. Locally constrained graph homomorphisms Jan Kratochvíl Jan Kratochvíl Charles University, Prague

2. Outline of the talk Graph homomorphism Graph homomorphism Local constraints - graph covers partial covers – frequency assignment role assignments Local constraints - graph covers partial covers – frequency assignment role assignments Complexity results and questions Complexity results and questions

3. 1. Graph homomorphism Edge preserving vertex mapping between graphs G and H f : V(G)  V(H) s.t. f : V(G)  V(H) s.t. uv  E(G)  f(u)f(v)  E(H) uv  E(G)  f(u)f(v)  E(H)

4. u v f(u)f(u) f(v)f(v) f G H

6. H-COLORING Input: A graph G. Question:  homomorphism G  H? Thm (Hell, Nešetřil): H-COLORING is polynomial for H bipartite and NP-complete otherwise.

7. 2. Local constraints For every u  V(G), f(N G (u))  N H (f(u)) f(N G (u))  N H (f(u))

8. u f f(u)f(u) G H

9. Definition: A homomorphism f : G  H is called bijective locally injective if for every u  V(G) surjective the restricted mapping f : N G (u))  N H (f(u)) bijective is injective. surjective

10. 2. Locally constrained homomorphisms loc. bijective = graph covers loc. bijective = graph covers loc. injective = partial covers = generalized frequency assignment loc. injective = partial covers = generalized frequency assignment loc. surjective = role assignment loc. surjective = role assignment computational complexity computational complexity

11. 2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Degree preserving Degree preserving

13. 2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Degree preserving Degree preserving Local computation (Angluin, Courcelle) Local computation (Angluin, Courcelle)

14. 2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Degree preserving Degree preserving Local computation (Angluin, Courcelle) Local computation (Angluin, Courcelle) Degree partition preserving Degree partition preserving

15. Degree partition – the coarsest partition V(G) = V 1  V 2  …  V t s.t. V(G) = V 1  V 2  …  V t s.t. there exist numbers r ij s.t. | N(v)  V j | = r ij for every v  V i. | N(v)  V j | = r ij for every v  V i.

25. 2.1 Locally bijective homomorphisms = graph covers Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) Degree preserving Degree preserving Local computation (Angluin, Courcelle) Local computation (Angluin, Courcelle) Degree partition preserving Degree partition preserving Finite planar covers Finite planar covers

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

29. Attempts to prove via forbidden minors for projective planar graphs (Negami, Fellows, Archdeacon, Hliněný)

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, Hliněný) True if K 1,2,2,2 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, Škrekovski)

34. L(2,1)-labelings of graphs f: V(G)  {0,1,2,…,k} uv  E(G)  |f(u) – f(v)|  2 uv  E(G)  |f(u) – f(v)|  2 d G (u,v) = 2  f(u)  f(v) d G (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 uv  E(G)  |f(u) – f(v)|  2 d G (u,v) = 2  f(u)  f(v) d G (u,v) = 2  f(u)  f(v) |f(u) – f(v)|  1 |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 uv  E(G)  |f(u) – f(v)|  2 d G (u,v) = 2  f(u)  f(v) d G (u,v) = 2  f(u)  f(v) |f(u) – f(v)|  1 |f(u) – f(v)|  1 L (2,1) (G) = min such k

43. L(2,1)-labelings of graphs NP-complete for every fixed k  4 (Fiala, Kloks, JK) NP-complete for every fixed k  4 (Fiala, Kloks, JK) Polynomial for graphs of bounded tree- width (when k fixed) 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) NP-complete for every fixed k  4 (Fiala, Kloks, JK) Polynomial for graphs of bounded tree-width (when k fixed) Polynomial for graphs of bounded tree-width (when k fixed) Polynomial for trees when k part of input (Chang, Kuo) Polynomial for trees when k part of input (Chang, Kuo) Open for graphs of bounded tree-width (when k part of input) 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) C k (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)  d H ( f(u), f(v))  2 uv  E(G)  d H ( f(u), f(v))  2 d G (u,v) = 2  f(u)  f(v) d G (u,v) = 2  f(u)  f(v)

48. H(2,1)-labelings of graphs f: V(G)  V(H) uv  E(G)  d H ( f(u), f(v))  2 uv  E(G)  d H ( f(u), f(v))  2  f(u)f(v)  E(H)  f(u)f(v)  E(H) d G (u,v) = 2  f(u)  f(v) d G (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) uv  E(G)  f(u)f(v)  E(-H) d G (u,v) = 2  f(u)  f(v) d G (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) uv  E(G)  f(u)f(v)  E(-H) homomorphism from G to -H homomorphism from G to -H d G (u,v) = 2  f(u)  f(v) d G (u,v) = 2  f(u)  f(v) locally injective locally injective

51. H(2,1)-labelings of graphs = locally injective homomorphisms into –H

52. L 2,1 (G)  k iff G allows a P k+1 (2,1)-labeling iff G allows a locally injective homomorphism into -P k+1.

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, Nešetřil): 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 Bodlaender 1989 Abello, Fellows, Stilwell 1991 Abello, Fellows, Stilwell 1991 JK, Proskurowski, Telle 1994, 1996, 1997 JK, Proskurowski, Telle 1994, 1996, 1997 Jiří Fiala 2000 Jiří Fiala 2000

58. Complexity of H-COVER NP-complete for k-regular graphs H (k  3) NP-complete for k-regular graphs H (k  3)

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

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

64. Theorem (KPT): G covers  (a 1 n 1,a 2 n 2,…,a k n k ) if and only if G contains only vertices of degrees 2 and d = n 1 + n 2 + … + n k, and the vertices of degree d can be colored by two colors red and blue so that each one is connected by exactly n i paths of length a i to the vertices of the opposite color.

65. G  (a 1 n 1,a 2 n 2,…,a k n k )

67. (aini)(aini)(aini)(aini) G’

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

71. W(a 1 n 1,a 2 n 2,…,a k n k ;a 1 l 1,a 2 l 2,…,a k l k ;a 1 m 1,a 2 m 2,…,a k m k )

72. Theorem (KPT): The W-COVER problem is NP-complete if n i = m i for all i, and n i = m i for all i, and n i. l i > 0 for some i n i. l i > 0 for some i and polynomial time solvable otherwise. 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. 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: C k (2,1)-labeling is NP-complete for every k  6.

77. Partial covers of Theta graphs

78. Thm (Fiala, JK):  (a k,b m )-PARTIAL-COVER is - polynomial if a,b are odd - polynomial if a,b are odd - NP-complete if a-b is odd - NP-complete if a-b is odd

79. Partial covers of Theta graphs Thm (FK):  (a k,b m )-PARTIAL-COVER is - polynomial if a,b are odd - polynomial if a,b are odd - NP-complete if a-b is 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 NP-complete if a,b,c are distinct odd integers

80. Partial covers of Theta graphs Thm:  (a k,b m )-PARTIAL-COVER is - polynomial if a,b are odd - polynomial if a,b are odd - NP-complete if a-b is 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 NP-complete if a,b,c are distinct odd integers Thm (FK):  (a,b,c)-PARTIAL-COVER is NP-complete if a+b|c NP-complete if a+b|c

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(1;1;1)-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(1;1;1)-COVER). Given G, construct G’ by replacing its edges by paths of length c.

85. c = a + b + a + b + … + a + b = b + a + b + a + … + b + a = b + a + b + a + … + b + a = c = c

86. G’

87. G

88. 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(1;1;1)-COVER). Construct G’ by replacing its edges by paths of length c. Then G’ partially covers  (a,b,c) iff G covers W(1;1;1). G covers W(1;1;1).

91.  (1,2,3)

93. -  (1,2,3) = P 5

94. Eq:  (1,2,3)-PARTIAL-COVER  (1,2,3)-PARTIAL-COVER P 5 (2,1)-labeling P 5 (2,1)-labeling L (2,1) (G)  4 L (2,1) (G)  4

95. Eq:  (1,2,3)-PARTIAL-COVER  (1,2,3)-PARTIAL-COVER P 5 (2,1)-labeling P 5 (2,1)-labeling L (2,1) (G)  4 L (2,1) (G)  4 And hence all NP-complete. And hence all NP-complete.

96. Questions – Partial cover More than 3 paths -  (a,b,c,d,…) More than 3 paths -  (a,b,c,d,…) Multiple lengths -  (a n,b m,c k ) Multiple lengths -  (a n,b m,c k )

97. Questions – Partial cover More than 3 paths -  (a,b,c,d,…) More than 3 paths -  (a,b,c,d,…) Multiple lengths -  (a n,b m,c k ) Multiple lengths -  (a n,b m,c k ) Beyond Theta graphs – Beyond Theta graphs – H-PARTIAL-COVER is conjectured H-PARTIAL-COVER is conjectured NP-complete for H containing a NP-complete for H containing a subdivision of K 4 subdivision of K 4

98. Questions – Partial cover Dichotomy ? Dichotomy ? Plausible conjecture ? Plausible conjecture ?

99. Questions – Cover Dichotomy ? Dichotomy ?

100. Questions – Cover Dichotomy ? Dichotomy ? Perhaps affine graphs and graphs with Unique Neighbor Property are the only polynomial cases for H-COVER 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 Planar instances

102. Thank you Thank you

103. 6th Czech-Slovak International Symposium on Graphs and Combinatorics Prague, July 10-15, 2006 In honor of 60 th birthday of Jarik Nešetřil