Advertisement

On Graphs Whose Maximal Cliques and Stable Sets Intersect

  • Diogo V. Andrade
  • Endre Boros
  • Vladimir GurvichEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 139)

Abstract

We say that a graph G has the CIS-property and call it a CIS-graph if every maximal clique and every maximal stable set of G intersects.

By definition, G is a CIS-graph if and only if the complementary graph \(\overline {G}\) is a CIS-graph. Let us substitute a vertex v of a graph G′ by a graph G″ and denote the obtained graph by G. It is also easy to see that G is a CIS-graph if and only if both G′ and G″ are CIS-graphs. In other words, CIS-graphs respect complementation and substitution. Yet, this class is not hereditary, that is, an induced subgraph of a CIS-graph may have no CIS-property. Perhaps, for this reason, the problems of efficient characterization and recognition of CIS-graphs are difficult and remain open. In this paper we only give some necessary and some sufficient conditions for the CIS-property to hold.

There are obvious sufficient conditions. It is known that P4-free graphs have the CIS-property and it is easy to see that G is a CIS-graph whenever each maximal clique of G has a simplicial vertex. However, these conditions are not necessary.

There are also simple necessary conditions. Given an integer k ≥ 2, a comb (or k-comb) Sk is a graph with 2k vertices k of which, v1, …, vk, form a clique C, while others, \(v^{\prime }_1, \ldots , v^{\prime }_k,\) form a stable set S, and \((v_i,v^{\prime }_i)\) is an edge for all i = 1, …, k, and there are no other edges. The complementary graph \(\overline {S_k}\) is called an anti-comb (or k-anti-comb). Clearly, S and C switch in the complementary graphs. Obviously, the combs and anti-combs are not CIS-graphs, since C ∩ S = ∅. Hence, if a CIS-graph G contains an induced comb or anti-comb, then it must be settled, that is, G must contain a vertex v connected to all vertices of C and to no vertex of S. For k = 2 this observation was made by Claude Berge in 1985. However, these conditions are only necessary.

The following sufficient conditions are more difficult to prove: G is a CIS-graph whenever G contains no induced 3-combs and 3-anti-combs, and every induced 2-comb is settled in G, as it was conjectured by Vasek Chvatal in early 90s. First partial results were published by his student Wenan Zang from Rutgers Center for Operations Research. Then, the statement was proven by Deng, Li, and Zang. Here we give an alternative proof, which is of independent interest; it is based on some properties of the product of two Petersen graphs.

It is an open question whether G is a CIS-graph if it contains no induced 4-combs and 4-anti-combs, and all induced 3-combs, 3-anti-combs, and 2-combs are settled in G.

We generalize the concept of CIS-graphs as follows. For an integer d ≥ 2 we define a d-graph \(\mathcal {G} = (V; E_1, \ldots , E_d)\) as a complete graph whose edges are colored by d colors (that is, partitioned into d sets). We say that \(\mathcal {G}\) is a CIS-d-graph (has the CIS-d-property) if \(\bigcap _{i=1}^d C_i \neq \emptyset \) whenever for each i = 1, …, d the set Ci is a maximal color i-free subset of V , that is, (v, v′)∉Ei for any v, v′∈ Ci. Clearly, in case d = 2 we return to the concept of CIS-graphs. (More accurately, CIS-2-graph is a pair of two complementary CIS-graphs.) We conjecture that each CIS-d-graph is a Gallai graph, that is, it contains no triangle colored by 3 distinct colors. We obtain results supporting this conjecture and also show that if it holds then characterization and recognition of CIS-d-graphs are easily reduced to characterization and recognition of CIS-graphs.

We also prove the following statement. Let \(\mathcal {G} = (V; E_1, \ldots , E_d)\) be a Gallai d-graph such that at least d − 1 of its d chromatic components are CIS-graphs, then \(\mathcal {G}\) has the CIS-d-property. In particular, the remaining chromatic component of \(\mathcal {G}\) is a CIS-graph too. Moreover, all 2d unions of d chromatic components of \(\mathcal {G}\) are CIS-graphs.

Notes

Acknowledgements

The third author was partially funded by the Russian Academic Excellence Project ‘5-100’.

References

  1. 1.
    R.N. Ball, A. Pultr, P. Vojtěchovský, Colored graphs without colorful cycles. Combinatorica 27(4), 407–427 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    C. Berge, Problems 9.11 and 9.12, in Graphs and Order, ed. by I. Rival (Reidel, Dordrecht, 1985), pp. 583–584Google Scholar
  3. 3.
    E. Boros, V. Gurvich, Perfect graphs are kernel-solvable. Discret. Math. 159, 35–55 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    E. Boros, V. Gurvich, Stable effectivity functions and perfect graphs. Math. Soc. Sci. 39, 175–194 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. Boros, V. Gurvich, A. Vasin, Stable families of coalitions and normal hypergraphs. Math. Soc. Sci. 34, 107–123 (1997). RUTCOR Research Report, RRR-22-1995, Rutgers UniversityMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    E. Boros, V. Gurvich, P.L. Hammer, Dual subimplicants of positive Boolean functions. Optim. Methods Softw. 10, 147–156 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. Boros, V. Gurvich, I. Zverovich, On split and almost CIS-graphs. Aust. J. Comb. 43, 163–180 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Brandstädt, V.B. Le, J.P. Spinrad, Graph Classes: A Survey (SIAM, Philadelphia, 1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    H. Buer, R. Möring, A fast algorithm for decomposition of graphs and posets. Math. Oper. Res. 3, 170–184 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    K. Cameron, J. Edmonds, Lambda composition. J. Graph Theory 26, 9–16 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    K. Cameron, J. Edmonds, L. Lovász, A note on perfect graphs. Period. Math. Hung. 17(3), 441–447 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    F.R.K. Chung, R.L. Graham, Edge-colored complete graphs with precisely colored subgraphs. Combinatorica 3, 315–324 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Cournier, M. Habib, A new linear algorithm for modular decomposition, in Proceedings of 19th International Colloquium on Trees in Algebra and Programming (CAAP-94), Edinburgh, ed. by S. Tison. Lecture Notes in Computer Science, vol. 787 (Springer, Berlin, 1994), pp. 68–82Google Scholar
  14. 14.
    X. Deng, G. Li, W. Zang, Proof of Chvatal’s conjecture on maximal stable sets and maximal cliques in graphs. J. Comb. Theory Ser. B 91(2), 301–325 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    X. Deng, G. Li, W. Zang, Corrigendum to proof of Chvatal’s conjecture on maximal stable sets and maximal cliques in graphs. J. Comb. Theory Ser. B 94, 352–353 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    T. Eiter, Exact transversal hypergraphs and application to Boolean μ-functions. J. Symb. Comput. 17, 215–225 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    T. Eiter, Generating Boolean μ-expressions. Acta Informatica 32, 171–187 (1995)MathSciNetzbMATHGoogle Scholar
  18. 18.
    P. Erdős, M. Simonovits, V.T. Sos, Anti-Ramsey theorems. Colloq. Math. Soc. Janos Bolyai 10, 633–643 (1973)Google Scholar
  19. 19.
    S. Foldes, P.L. Hammer, Split graphs, in Proceedings of the 8th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State University, Baton Rouge, LA, 1977). Congressus Numerantium, vol. XIX (Utilitas Mathematica Publisher, Winnipeg, 1977), pp. 311–315Google Scholar
  20. 20.
    T. Gallai, Transitiv orientierbare graphen. Acta Math. Acad. Sci. Hungar. 18(1–2), 25–66 (1967). English translation by F. Maffray, M. Preissmann, Chapter 3: Perfect graphs, ed. by J.L.R. Alfonsin, B.A. Reed (Wiley, Hoboken, 2001)Google Scholar
  21. 21.
    M.C. Golumbic, V. Gurvich, Read-once Boolean functions, in Boolean Functions: Theory, Algorithms, and Applications, ed. by Y. Crama, P.L. Hammer (Cambridge University Press, Cambridge, 2011), pp. 448–486CrossRefGoogle Scholar
  22. 22.
    V. Gurvich, On repetition-free Boolean functions. Russ. Math. Surv. 32(1), 183–184 (1977) (in Russian)MathSciNetzbMATHGoogle Scholar
  23. 23.
    V. Gurvich, Applications of Boolean functions and contact schemes in game theory, section 5, Repetition-free Boolean functions and normal forms of positional games, Ph.D. thesis, Moscow Institute of Physics and Technology, Moscow, USSR (in Russian), 1978Google Scholar
  24. 24.
    V. Gurvich, On the normal form of positional games. Soviet Math. Dokl. 25(3), 572–575 (1982)zbMATHGoogle Scholar
  25. 25.
    V. Gurvich, Some properties and applications of complete edge-chromatic graphs and hypergraphs. Soviet Math. Dokl. 30(3), 803–807 (1984)zbMATHGoogle Scholar
  26. 26.
    V. Gurvich, Criteria for repetition-freeness of functions in the Algebra of Logic. Russ. Acad. Sci. Dokl. Math. 43(3), 721–726 (1991)zbMATHGoogle Scholar
  27. 27.
    V. Gurvich, Positional game forms and edge-chromatic graphs. Russ. Acad. Sci. Dokl. Math. 45(1), 168–172 (1992)Google Scholar
  28. 28.
    V. Gurvich, Decomposing complete edge-chromatic graphs and hypergraphs, revisited. Discret. Appl. Math. 157, 3069–3085 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. Gyárfás, G. Simonyi, Edge coloring of complete graphs without tricolored triangles. J. Graph Theory 46, 211–216 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    M. Karchmer, N. Linial, L. Newman, M. Saks, A. Wigderson, Combinatorial characterization of read-once formulae. Discret. Math. 114, 275–282 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    T. Kloks, C.-M. Lee, J. Liu, H. Müller, On the recognition of general partition graphs, in Graph-Theoretic Concepts of Computer Science. Lecture Notes in Computer Science, vol. 2880 (Springer, Berlin, 2003), pp. 273–283CrossRefGoogle Scholar
  32. 32.
    J. Körner, G. Simonyi, Graph pairs and their entropies: modularity problems. Combinatorica 20, 227–240 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J. Körner, G. Simonyi, Zs. Tuza, Perfect couples of graphs. Combinatorica 12, 179–192 (1992)Google Scholar
  34. 34.
    L. Lovász, Normal hypergraphs and the weak perfect graph conjecture. Discret. Math. 2(3), 253–267 (1972)CrossRefzbMATHGoogle Scholar
  35. 35.
    L. Lovász, A characterization of perfect graphs. J. Comb. Theory Ser. B 13(2), 95–98 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    L. Lovász, Combinatorial Problems and Exercises (North-Holland Publishing, Amsterdam, 1979)zbMATHGoogle Scholar
  37. 37.
    K. McAvaney, J. Robertson, D. DeTemple, A characterization and hereditary properties for partition graphs. Discret. Math. 113(1–3), 131–142 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    R.M. McCollel, J.P. Spinrad, Modular decomposition and transitive orientation. Discret. Math. 201, 189–241 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    R. Möring, Algorithmic aspects of the substitution decomposition in optimization over relations, set systems, and Boolean functions. Ann. Oper. Res. 4, 195–225 (1985/1986)MathSciNetCrossRefGoogle Scholar
  40. 40.
    J. Muller, J. Spinrad, Incremental modular decomposition. J. ACM 36(1), 1–19 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Yu.L. Orlovich, I.E. Zverovich, Independent domination and the triangle condition. Electron. Notes Discrete Math. 28, 341–348 (2007)CrossRefzbMATHGoogle Scholar
  42. 42.
    G. Ravindra, Strongly perfect line graphs and total graphs, in Finite and Infinite Sets, 6-th Hungarian Combinatorial Colloquium, vol. 2, Eger, 1981. Colloquia Mathematica Societatis Janos Bolyai, vol. 37 (North Holland, Amsterdam, 1984) 621–633.Google Scholar
  43. 43.
    Y. Wu, W. Zang, C.-Q. Zhang, A characterization of almost CIS graphs. SIAM J. Discret. Math. 23(2), 749–753 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    W. Zang, Generalizations of Grillet’s theorem on maximal stable sets and maximal cliques in graphs. Discret. Math. 143, 259–268 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    I. Zverovich, I. Zverovich, Bipartite hypergraphs: a survey and new results. Discret. Math. 306, 801–811 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Diogo V. Andrade
    • 1
  • Endre Boros
    • 2
  • Vladimir Gurvich
    • 3
    Email author
  1. 1.Google Inc NYCNew YorkUSA
  2. 2.RUTCOR, Rutgers UniversityPiscatawayUSA
  3. 3.National Research University Higher School of Economics (HSE)MoscowRussia

Personalised recommendations