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Upper Clique Transversals in Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2023)

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Abstract

A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the “upper” variant of this parameter, the upper clique transversal number, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs and proper interval graphs.

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References

  1. Abbas, N., Stewart, L.K.: Biconvex graphs: ordering and algorithms. Discrete Appl. Math. 103(1–3), 1–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. AbouEisha, H., Hussain, S., Lozin, V., Monnot, J., Ries, B., Zamaraev, V.: Upper domination: towards a dichotomy through boundary properties. Algorithmica 80(10), 2799–2817 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andreae, T., Flotow, C.: On covering all cliques of a chordal graph. Discrete Math. 149(1–3), 299–302 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreae, T., Schughart, M., Tuza, Z.: Clique-transversal sets of line graphs and complements of line graphs. Discrete Math. 88(1), 11–20 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balachandran, V., Nagavamsi, P., Rangan, C.P.: Clique transversal and clique independence on comparability graphs. Inform. Process. Lett. 58(4), 181–184 (1996)

    Article  MathSciNet  Google Scholar 

  6. Bazgan, C., et al.: The many facets of upper domination. Theoret. Comput. Sci. 717, 2–25 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berry, A., Pogorelcnik, R.: A simple algorithm to generate the minimal separators and the maximal cliques of a chordal graph. Inform. Process. Lett. 111(11), 508–511 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bertossi, A.A.: Dominating sets for split and bipartite graphs. Inform. Process. Lett. 19(1), 37–40 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Beyer, T., Proskurowski, A., Hedetniemi, S., Mitchell, S.: Independent domination in trees. In: Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), pp. 321–328. Congressus Numerantium, No. XIX (1977)

    Google Scholar 

  10. Bonomo, F., Durán, G., Safe, M.D., Wagler, A.K.: Clique-perfectness of complements of line graphs. Discrete Appl. Math. 186, 19–44 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boria, N., Della Croce, F., Paschos, V.T.: On the MAX MIN VERTEX COVER problem. Discrete Appl. Math. 196, 62–71 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. In: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, SIAM Monographs on Discrete Mathematics and Applications (1999)

    Google Scholar 

  13. Chang, G.J., Farber, M., Tuza, Z.: Algorithmic aspects of neighborhood numbers. SIAM J. Discrete Math. 6(1), 24–29 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chang, J.M.: Induced matchings in asteroidal triple-free graphs. Discrete Appl. Math. 132(1–3), 67–78 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cooper, J.W., Grzesik, A., Král, D.: Optimal-size clique transversals in chordal graphs. J. Graph Theory 89(4), 479–493 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Damaschke, P.: Parameterized algorithms for double hypergraph dualization with rank limitation and maximum minimal vertex cover. Discrete Optim. 8(1), 18–24 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Damaschke, P., Müller, H., Kratsch, D.: Domination in convex and chordal bipartite graphs. Inform. Process. Lett. 36(5), 231–236 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dublois, L., Hanaka, T., Khosravian Ghadikolaei, M., Lampis, M., Melissinos, N.: (In) approximability of maximum minimal FVS. J. Comput. System Sci. 124, 26–40 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eades, P., Keil, M., Manuel, P.D., Miller, M.: Two minimum dominating sets with minimum intersection in chordal graphs. Nordic J. Comput. 3(3), 220–237 (1996)

    MathSciNet  Google Scholar 

  20. Erdős, P., Gallai, T., Tuza, Z.: Covering the cliques of a graph with vertices. Discrete Math. 108, 279–289 (1992)

    Google Scholar 

  21. Ferneyhough, S., Haas, R., Hanson, D., MacGillivray, G.: Star forests, dominating sets and Ramsey-type problems. Discrete Math. 245(1–3), 255–262 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gardi, F.: The Roberts characterization of proper and unit interval graphs. Discrete Math. 307(22), 2906–2908 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guruswami, V., Pandu Rangan, C.: Algorithmic aspects of clique-transversal and clique-independent sets. Discrete Appl. Math. 100(3), 183–202 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hedetniemi, S.T.: A max-min relationship between matchings and domination in graphs. Congr. Numer. 40, 23–34 (1983)

    MathSciNet  MATH  Google Scholar 

  25. Jacobson, M.S., Peters, K.: Chordal graphs and upper irredundance, upper domination and independence. Discrete Math. 86(1–3), 59–69 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khoshkhah, K., Ghadikolaei, M.K., Monnot, J., Sikora, F.: Weighted upper edge cover: complexity and approximability. J. Graph Algorithms Appl. 24(2), 65–88 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lampis, M., Melissinos, N., Vasilakis, M.: Parameterized max min feedback vertex set. In: Proceedings of 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023) (2023). arXiv:2302.09604

  28. Lee, C.M.: Algorithmic aspects of some variations of clique transversal and clique independent sets on graphs. Algorithms (Basel) 14(1), 22 (2021)

    Article  MathSciNet  Google Scholar 

  29. Lee, C.M., Chang, M.S.: Distance-hereditary graphs are clique-perfect. Discrete Appl. Math. 154(3), 525–536 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lin, M.C., Vasiliev, S.: Approximation algorithms for clique transversals on some graph classes. Inform. Process. Lett. 115(9), 667–670 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, K., Lu, M.: Complete-subgraph-transversal-sets problem on bounded treewidth graphs. J. Comb. Optim. 41(4), 923–933 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  32. Milanič, M.: Strong cliques and stable sets. In: Topics in algorithmic graph theory, Encyclopedia of Mathematics and its Applications, vol. 178, pp. 207–227. Cambridge University Press, Cambridge (2021)

    Google Scholar 

  33. Monnot, J., Fernau, H., Manlove, D.: Algorithmic aspects of upper edge domination. Theoret. Comput. Sci. 877, 46–57 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Müller, H., Brandstädt, A.: The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs. Theoret. Comput. Sci. 53(2–3), 257–265 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nguyen, C.T., Shen, J., Hou, M., Sheng, L., Miller, W., Zhang, L.: Approximating the spanning star forest problem and its application to genomic sequence alignment. SIAM J. Comput. 38(3), 946–962 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Payan, C.: Remarks on cliques and dominating sets in graphs. Ars Combin. 7, 181–189 (1979)

    MathSciNet  MATH  Google Scholar 

  37. Shan, E., Liang, Z., Kang, L.: Clique-transversal sets and clique-coloring in planar graphs. Eur. J. Combin. 36, 367–376 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  39. West, D.B.: Introduction to graph theory. Prentice Hall Inc, Upper Saddle River, NJ (1996)

    Google Scholar 

  40. Zang, W.: Generalizations of Grillet’s theorem on maximal stable sets and maximal cliques in graphs. Discrete Math. 143(1–3), 259–268 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zehavi, M.: Maximum minimal vertex cover parameterized by vertex cover. SIAM J. Discrete Math. 31(4), 2440–2456 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are grateful to Nikolaos Melissinos, Haiko Müller, and the anonymous reviewers for their helpful comments. The work of the first named author is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0102, N1-0160, J1-3001, J1-3002, J1-3003, J1-4008, and J1-4084). Part of the work was done while the author was visiting Osaka Prefecture University in Japan, under the operation Mobility of Slovene higher education teachers 2018–2021, co-financed by the Republic of Slovenia and the European Union under the European Social Fund. The second named author is partially supported by JSPS KAKENHI Grant Number JP17K00017, 20H05964, and 21K11757, Japan.

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Authors and Affiliations

  1. FAMNIT and IAM, University of Primorska, Koper, Slovenia

    Martin Milanič

  2. Graduate School of Informatics, Osaka Metropolitan University, Sakai, Japan

    Yushi Uno

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  1. Martin Milanič
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  2. Yushi Uno
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Correspondence to Martin Milanič .

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Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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Milanič, M., Uno, Y. (2023). Upper Clique Transversals in Graphs. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_31

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  • DOI: https://doi.org/10.1007/978-3-031-43380-1_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43379-5

  • Online ISBN: 978-3-031-43380-1

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