- Email: [email protected]
On the structure of self-complementary graphs
Electronic Notes in Discrete Mathematics 22 (2005) 79–82 www.elsevier.com/locate/endm
On the structure of self-complementary graphs Nicolas Trotignon 1 Laboratoire Leibniz-IMAG Grenoble, France
Keywords: graph, self-complementary, decomposition.
1
Introduction
Here graphs are simple, non-oriented, with no loop and finite. A graph is selfcomplementary if G is isomorphic to its complement G. We refer the reader to a survey on self-complementary graphs due to Farrugia [9]. We write “scgraph” for “self-complementary graph”. Colbourn et al. [5] proved that a polynomial time recognition algorithm of sc-graphs would imply a polynomial time algorithm for the isomorphism problem. We aim at structural properties of sc-graphs, saying something like: every sc-graph either contains some prescribed induced subgraph or can be partitioned into sets of vertices with some prescribed adjacencies. Farrugia [9] mentions only one such theorem: Theorem 1.1 (Gibbs, [11]) An sc-graph on 4k vertices contains k disjoint induced P4 ’s. This theorem has two major defaults in view of algorithms. The problem of deciding if a graph can be partitioned into sets of 4 vertices inducing P4 ’s 1
Email: [email protected]
1571-0653/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.06.014
80
N. Trotignon / Electronic Notes in Discrete Mathematics 22 (2005) 79–82
is NP-complete (Kirkpatrick and Hell, [12]). Moreover even if the partition of an sc-graph is given, it is useless for recursion: removing one of the P4 ’s may yield a graph with no forseeable properties. We investigate structural properties of sc-graphs fixing the first default but still having the second one.
2
A new proof of Gibb’s theorem
If G is a graph, we denote by V (G) the vertex set of G, by E(G) the edge set of G. If A ⊂ V (G), we denote by G[A] the subgraph of G induced by A. A graph G is self-complementary if and only if there exists a bijection τ from V (G) to V (G) such that for every pair {a, b} of distinct vertices we have: {a, b} ∈ E(G) ⇔ {τ (a), τ (b)} ∈ / E(G). Such a function τ is called an antimorphism of G. Sachs [14] and Ringel [13] proved that any antimorphism is a product of circular permutations whose lengths are all multiples of 4, except possibly for one of length 1. It is convenient to denote by (a1 a2 . . . ak ) the circular permutation of {a1 , a2 , . . . , ak } that maps ai to ai+1 , where the addition of the subscripts is taken modulo k. When a circular permutation has length 4k, we denote it by (a1 b1 c1 d1 a2 b2 c2 d2 . . . ak bk ck dk ). Implicitly, the subscripts are then taken modulo k (for instance ak+3 = a3 , d0 = dk , . . . ). The following lemma is used in the proof of Theorem 3.1 and allows us to prove again Theorem 1.1. A symmetric partition in a graph G is a partition (A, B, C, D) of V (G) such that each of A, B, C, D is non-empty, there are no edges between A, D, no edges between B, C, every possible edges between A, B, and every possible edges between C, D. Lemma 2.1 Let k ≥ 1 be an integer and G be an sc-graph with an antimorphism τ = (a1 b1 c1 d1 a2 b2 c2 d2 . . . ak bk ck dk )(. . .) · · · (. . .). Put A = {a1 , . . . , ak }, B = {b1 , . . . , bk }, C = {c1 , . . . , ck }, D = {d1 , . . . , dk }. Then either there exists i, j ∈ N such that {a1 , bi , a1+j , bi+j } induces a P4 for which (a1 bi a1+j bi+j ) is an antimorphism, or one of (A, B, C, D), (B, C, D, A) is a symmetric partition of G[A ∪ B ∪ C ∪ D].
3
A structural theorem and a conjecture
A skew partition in a graph G is a partition (A, B, C, D) of V (G) such that each of A, B, C, D is non-empty, there are no edges between A, B and every possible edges between C, D. Theorem 3.1 Let G be an sc-graph with an antimorphism τ that is the product of two circular permutations, one of them of length 4. Then either G
N. Trotignon / Electronic Notes in Discrete Mathematics 22 (2005) 79–82
81
contains a C5 as an induced subgraph, or G contains a skew partition, or G contains a symmetric partition. Let us now discuss the motivivation and possible extensions of Theorem 3.1. Skew partitions were introduced by Chv´atal for the study of perfect graphs [4], and play an important role in the proof of Strong Perfect Graph Conjecture by Chudnovsky, Robertson Seymour and Thomas [3]. Symmetric partitions may be seen as a very particular case of the 2-join defined by Cunningham and Cornu´ejols, once again for the study of perfect graphs [6]. A 2-join in G is a partition (X1 , X2 ) of V (G) so that there exist disjoint non-empty Ai , Bi ⊂ Xi , (i = 1, 2) satisfaying: (i) Every vertex of A1 is adjacent to every vertex of A2 , every vertex of B1 is adjacent to every vertex of B2 , and there are no other edges between X1 and X2 ; (ii) For i = 1, 2, every component of G[Xi ] meets both Ai and Bi . For i = 1, 2, if |Ai | = |Bi | = 1, and if Xi induces a path of G joining the vertex of Ai and the vertex of Bi , then it has length at least 3. The conditions in (2), known as the technical requirements, are important for applications to perfect graphs. If a graph G has a 2-join, then (A1 , A2 , B1 , B2 ) is a symmetric partition of G[A1 ∪ A2 ∪ B1 ∪ B2 ]. So, if we forget the technical requirements, symmetric partitions are special 2-joins (such that Xi \ (Ai ∪ Bi ) = ∅, i = 1, 2). A lot of work has been done on finding algorithms that decide if the a graph can be partitioned into several subgraphs with restrictions on the adjacencies [1,7,10]). Symmetric partitions are detectable in linear time [7]. Figueiredo, Klein, Kohayakawa and Reed gave a polynomial time algorithm that decides whether a graph has a skew partition [8]. So, the outcomes of Theorem 3.1 are testable in polynomial time. We conjecture the following: Conjecture 3.2 Conclusion of Theorem 3.1 holds for any sc-graph on 4k vertices. This conjecture has an analogy with the theorem of Chudnovsky, Robertson, Seymour and Thomas for decomposing Berge Graphs. It would be nice to have a stronger theorem in the particular case of Berge sc-graphs. Conjecture 3.2 could be a candidate. A hole in a graph is an induced cycle of length at least 4. A graph is Berge if in both G, G, there is no hole of odd length. Theorem 3.3 (Chudnovsky et al.[2,3]) Let G be a Berge graph. Then either one of G, G is bipartite, or one of G, G is the line-graph of a bipartite
82
N. Trotignon / Electronic Notes in Discrete Mathematics 22 (2005) 79–82
graph, or one of G, G has a 2-join, or G has a skew partition.
References [1] K. Cameron, E. M. Eschen, C. T. Ho` ang, and R. Sritharan, The list partition problem for graphs, Proceedings of the ACM-SIAM Symposium on Discrete Algorithms - SODA 2004, ACM, New York and SIAM, Philadelphia, 2004, pp. 384–392. [2] M. Chudnovsky, Berge trigraphs, Manuscript, 2004. [3] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Annals of Math. (2002), To appear. [4] V. Chv´ atal, Star-cutsets and perfect graphs, J. Combin. Ser. B 39 (1985), 189– 199. [5] M.J. Colbourn and C.J. Colbourn, Graph isomorphism and self-complementary graphs, SIGACT News 10 (1978), 25–29. [6] G. Cornu´ejols and W. H. Cunningham, Composition for perfect graphs, Disc. Math. 55 (1985), 245–254. [7] S. Dantas, C. M. H. de Figueiredo, S. Gravier, and S. Klein, Finding Hpartitions efficiently, Manuscript, 2004. [8] C. M. H. de Figueiredo, S. Klein, Y. Kohayakawa, and B. Reed, Finding skew partitions efficiently, J. Algorithms 37 (2000), 505–521. [9] A. Farrugia, Self-complementary graphs and generalisations: a comprehensive reference manual, Master’s thesis, University of Malta, 1999. [10] T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition problems, Proceedings of the 31st annual ACM Symposium on Theory of Computing - STOC’99 (New York), Plenum Press, 1999, pp. 464–472. [11] R. A. Gibbs, Self-complementary graphs, Jour. Comb. Theory Ser. B 16 (1974), 106–123. [12] D. G. Kirkpatrick and P. Hell, On the complexity of a generalized matching problem, Proc. 10th Ann. ACM Symp. on Theory of Computing (New York), Association for Computing Machinery, 1978, pp. 240–245. [13] G. Ringel, Selbstkomplementare Graphen, Arch. Math. 14 (1963), 354–358. ¨ [14] H. Sachs, Uber Selbstkomplementare Graphen, Publ. Math. Debrecen 9 (1962), 270–288.
On the structure of self-complementary graphs Nicolas Trotignon 1 Laboratoire Leibniz-IMAG Grenoble, France
Keywords: graph, self-complementary, decomposition.
1
Introduction
Here graphs are simple, non-oriented, with no loop and finite. A graph is selfcomplementary if G is isomorphic to its complement G. We refer the reader to a survey on self-complementary graphs due to Farrugia [9]. We write “scgraph” for “self-complementary graph”. Colbourn et al. [5] proved that a polynomial time recognition algorithm of sc-graphs would imply a polynomial time algorithm for the isomorphism problem. We aim at structural properties of sc-graphs, saying something like: every sc-graph either contains some prescribed induced subgraph or can be partitioned into sets of vertices with some prescribed adjacencies. Farrugia [9] mentions only one such theorem: Theorem 1.1 (Gibbs, [11]) An sc-graph on 4k vertices contains k disjoint induced P4 ’s. This theorem has two major defaults in view of algorithms. The problem of deciding if a graph can be partitioned into sets of 4 vertices inducing P4 ’s 1
Email: [email protected]
1571-0653/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.06.014
80
N. Trotignon / Electronic Notes in Discrete Mathematics 22 (2005) 79–82
is NP-complete (Kirkpatrick and Hell, [12]). Moreover even if the partition of an sc-graph is given, it is useless for recursion: removing one of the P4 ’s may yield a graph with no forseeable properties. We investigate structural properties of sc-graphs fixing the first default but still having the second one.
2
A new proof of Gibb’s theorem
If G is a graph, we denote by V (G) the vertex set of G, by E(G) the edge set of G. If A ⊂ V (G), we denote by G[A] the subgraph of G induced by A. A graph G is self-complementary if and only if there exists a bijection τ from V (G) to V (G) such that for every pair {a, b} of distinct vertices we have: {a, b} ∈ E(G) ⇔ {τ (a), τ (b)} ∈ / E(G). Such a function τ is called an antimorphism of G. Sachs [14] and Ringel [13] proved that any antimorphism is a product of circular permutations whose lengths are all multiples of 4, except possibly for one of length 1. It is convenient to denote by (a1 a2 . . . ak ) the circular permutation of {a1 , a2 , . . . , ak } that maps ai to ai+1 , where the addition of the subscripts is taken modulo k. When a circular permutation has length 4k, we denote it by (a1 b1 c1 d1 a2 b2 c2 d2 . . . ak bk ck dk ). Implicitly, the subscripts are then taken modulo k (for instance ak+3 = a3 , d0 = dk , . . . ). The following lemma is used in the proof of Theorem 3.1 and allows us to prove again Theorem 1.1. A symmetric partition in a graph G is a partition (A, B, C, D) of V (G) such that each of A, B, C, D is non-empty, there are no edges between A, D, no edges between B, C, every possible edges between A, B, and every possible edges between C, D. Lemma 2.1 Let k ≥ 1 be an integer and G be an sc-graph with an antimorphism τ = (a1 b1 c1 d1 a2 b2 c2 d2 . . . ak bk ck dk )(. . .) · · · (. . .). Put A = {a1 , . . . , ak }, B = {b1 , . . . , bk }, C = {c1 , . . . , ck }, D = {d1 , . . . , dk }. Then either there exists i, j ∈ N such that {a1 , bi , a1+j , bi+j } induces a P4 for which (a1 bi a1+j bi+j ) is an antimorphism, or one of (A, B, C, D), (B, C, D, A) is a symmetric partition of G[A ∪ B ∪ C ∪ D].
3
A structural theorem and a conjecture
A skew partition in a graph G is a partition (A, B, C, D) of V (G) such that each of A, B, C, D is non-empty, there are no edges between A, B and every possible edges between C, D. Theorem 3.1 Let G be an sc-graph with an antimorphism τ that is the product of two circular permutations, one of them of length 4. Then either G
N. Trotignon / Electronic Notes in Discrete Mathematics 22 (2005) 79–82
81
contains a C5 as an induced subgraph, or G contains a skew partition, or G contains a symmetric partition. Let us now discuss the motivivation and possible extensions of Theorem 3.1. Skew partitions were introduced by Chv´atal for the study of perfect graphs [4], and play an important role in the proof of Strong Perfect Graph Conjecture by Chudnovsky, Robertson Seymour and Thomas [3]. Symmetric partitions may be seen as a very particular case of the 2-join defined by Cunningham and Cornu´ejols, once again for the study of perfect graphs [6]. A 2-join in G is a partition (X1 , X2 ) of V (G) so that there exist disjoint non-empty Ai , Bi ⊂ Xi , (i = 1, 2) satisfaying: (i) Every vertex of A1 is adjacent to every vertex of A2 , every vertex of B1 is adjacent to every vertex of B2 , and there are no other edges between X1 and X2 ; (ii) For i = 1, 2, every component of G[Xi ] meets both Ai and Bi . For i = 1, 2, if |Ai | = |Bi | = 1, and if Xi induces a path of G joining the vertex of Ai and the vertex of Bi , then it has length at least 3. The conditions in (2), known as the technical requirements, are important for applications to perfect graphs. If a graph G has a 2-join, then (A1 , A2 , B1 , B2 ) is a symmetric partition of G[A1 ∪ A2 ∪ B1 ∪ B2 ]. So, if we forget the technical requirements, symmetric partitions are special 2-joins (such that Xi \ (Ai ∪ Bi ) = ∅, i = 1, 2). A lot of work has been done on finding algorithms that decide if the a graph can be partitioned into several subgraphs with restrictions on the adjacencies [1,7,10]). Symmetric partitions are detectable in linear time [7]. Figueiredo, Klein, Kohayakawa and Reed gave a polynomial time algorithm that decides whether a graph has a skew partition [8]. So, the outcomes of Theorem 3.1 are testable in polynomial time. We conjecture the following: Conjecture 3.2 Conclusion of Theorem 3.1 holds for any sc-graph on 4k vertices. This conjecture has an analogy with the theorem of Chudnovsky, Robertson, Seymour and Thomas for decomposing Berge Graphs. It would be nice to have a stronger theorem in the particular case of Berge sc-graphs. Conjecture 3.2 could be a candidate. A hole in a graph is an induced cycle of length at least 4. A graph is Berge if in both G, G, there is no hole of odd length. Theorem 3.3 (Chudnovsky et al.[2,3]) Let G be a Berge graph. Then either one of G, G is bipartite, or one of G, G is the line-graph of a bipartite
82
N. Trotignon / Electronic Notes in Discrete Mathematics 22 (2005) 79–82
graph, or one of G, G has a 2-join, or G has a skew partition.
References [1] K. Cameron, E. M. Eschen, C. T. Ho` ang, and R. Sritharan, The list partition problem for graphs, Proceedings of the ACM-SIAM Symposium on Discrete Algorithms - SODA 2004, ACM, New York and SIAM, Philadelphia, 2004, pp. 384–392. [2] M. Chudnovsky, Berge trigraphs, Manuscript, 2004. [3] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Annals of Math. (2002), To appear. [4] V. Chv´ atal, Star-cutsets and perfect graphs, J. Combin. Ser. B 39 (1985), 189– 199. [5] M.J. Colbourn and C.J. Colbourn, Graph isomorphism and self-complementary graphs, SIGACT News 10 (1978), 25–29. [6] G. Cornu´ejols and W. H. Cunningham, Composition for perfect graphs, Disc. Math. 55 (1985), 245–254. [7] S. Dantas, C. M. H. de Figueiredo, S. Gravier, and S. Klein, Finding Hpartitions efficiently, Manuscript, 2004. [8] C. M. H. de Figueiredo, S. Klein, Y. Kohayakawa, and B. Reed, Finding skew partitions efficiently, J. Algorithms 37 (2000), 505–521. [9] A. Farrugia, Self-complementary graphs and generalisations: a comprehensive reference manual, Master’s thesis, University of Malta, 1999. [10] T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition problems, Proceedings of the 31st annual ACM Symposium on Theory of Computing - STOC’99 (New York), Plenum Press, 1999, pp. 464–472. [11] R. A. Gibbs, Self-complementary graphs, Jour. Comb. Theory Ser. B 16 (1974), 106–123. [12] D. G. Kirkpatrick and P. Hell, On the complexity of a generalized matching problem, Proc. 10th Ann. ACM Symp. on Theory of Computing (New York), Association for Computing Machinery, 1978, pp. 240–245. [13] G. Ringel, Selbstkomplementare Graphen, Arch. Math. 14 (1963), 354–358. ¨ [14] H. Sachs, Uber Selbstkomplementare Graphen, Publ. Math. Debrecen 9 (1962), 270–288.