Abstract
Domination is one of the classical subjects in structural graph theory and in graph algorithms. The Minimum Dominating Set problem and many of its variants are NP-complete and have been studied from various algorithmic perspectives. One of those variants called irredundance is highly related to domination. For example, every minimal dominating set of a graph G is also a maximal irredundant set of G. In this paper we study the enumeration of the maximal irredundant sets of a claw-free graph. We show that an n-vertex claw-free graph has \(O(1.9341^n)\) maximal irredundant sets and these sets can be enumerated in the same time. We complement the aforementioned upper bound with a lower bound by providing a family of graphs having \(1.5848^n\) maximal irredundant sets.
This work is supported by the Research Council of Norway (the project CLASSIS) and the French National Research Agency (ANR project GraphEn/ANR-15-CE40-0009).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abu-Khzam, F.N., Heggernes, P.: Enumerating minimal dominating sets in chordal graphs. Inf. Process. Lett. 116(12), 739–743 (2016)
Binkele-Raible, D., Brankovic, L., Cygan, M., Fernau, H., Kneis, J., Kratsch, D., Langer, A., Liedloff, M., Pilipczuk, M., Rossmanith, P., Wojtaszczyk, J.O.: Breaking the \(2^n\)-barrier for Irredundance: two lines of attack. J. Discret. Algorithms 9(3), 214–230 (2011)
Boral, A., Cygan, M., Kociumaka, T., Pilipczuk, M.: A fast branching algorithm for cluster vertex deletion. Theory Comput. Syst. 58(2), 357–376 (2016)
Chudnovsky, M., Seymour, P.: Claw-free graphs. V. Global structure. J. Comb. Theory Ser. B 98, 1373–1410 (2008)
Cockayne, E., Favaron, O., Payan, C., Thomason, A.: Contributions to the theory of domination, independence and irredundance in graphs. Discret. Math. 33, 249–258 (1981)
Couturier, J.F., Heggernes, P., van’t Hof, P., Kratsch, D.: Minimal dominating sets in graph classes: combinatorial bounds and enumeration. Theoret. Comput. Sci. 487, 82–94 (2013)
Couturier, J.-F., Letourneur, R., Liedloff, M.: On the number of minimal dominating sets on some graph classes. Theoret. Comput. Sci. 562, 634–642 (2015)
Fomin, F.V., Gaspers, S., Lokshtanov, D., Saurabh, S.: Exact algorithms via monotone local search. In: STOC 2016, pp. 764–775 (2016)
Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1), 9:1–9:17 (2008)
Fomin, F.V., Kratsch, D.: Exact exponential algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2010)
Fellows, M.R., Fricke, G., Hedetniemi, S.T., Jacobs, D.P.: The private neighbor cube. SIAM J. Discret. Math. 7, 41–47 (1994)
Golovach, P.A., Heggernes, P., Kanté, M.M., Kratsch, D., Villanger, Y.: Enumerating minimal dominating sets in chordal bipartite graphs. Discret. Appl. Math. 199, 30–36 (2016)
Golovach, P.A., Heggernes, P., Kratsch, D., Villanger, Y.: An incremental polynomial time algorithm to enumerate all minimal edge dominating sets. Algorithmica 72, 836–859 (2015)
Haynes, T.W., Hedetniemi, S.T., Slater, P.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1997)
Haynes, T.W., Hedetniemi, S.T., Slater, P. (eds.): Fundamentals of Domination in Graphs. CRC Press, New York (1998)
Hedetniemi, S.T., Laskar, R., Pfaff, J.: Irredundance in graphs: a survey. Congr. Numerantium 48, 183–193 (1985)
Jacobson, M.S., Peters, K.: Chordal graphs and upper irredundance, upper domination and independence. Discret. Math. 86, 59–69 (1990)
Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: On the neighbourhood helly of some graph classes and applications to the enumeration of minimal dominating sets. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 289–298. Springer, Heidelberg (2012). doi:10.1007/978-3-642-35261-4_32
Kanté, M.M., Limouzy, V., Mary, A., Nourine, L.: On the enumeration of minimal dominating sets and related notions. SIAM J. Discret. Math. 28(4), 1916–1929 (2014)
Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: On the enumeration and counting of minimal dominating sets in interval and permutation graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 339–349. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45030-3_32
Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: Polynomial delay algorithm for listing minimal edge dominating sets in graphs. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 446–457. Springer, Cham (2015). doi:10.1007/978-3-319-21840-3_37
Krzywkowski, M.: Trees having many minimal dominating sets. Inform. Proc. Lett. 113, 276–279 (2013)
Laskar, R., Pfaff, J.: Domination and irredundance in graphs, Technical report 434, Clemson University, Department of Mathematical (1983)
Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory Ser. B 28, 284–304 (1980)
Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discret. Math. 29, 53–76 (1980)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Golovach, P.A., Kratsch, D., Sayadi, M.Y. (2017). Enumeration of Maximal Irredundant Sets for Claw-Free Graphs. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-57586-5_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-57585-8
Online ISBN: 978-3-319-57586-5
eBook Packages: Computer ScienceComputer Science (R0)