Abstract
We consider the maximum independent set (MIS) problem, i.e., the problem asking for a vertex subset of maximum cardinality of a graph such that no two vertices in this set are adjacent. The problem is known to be NP-hard in general, even if restricted on graphs of maximum degree at most Δ for a given integer Δ ≥ 3, i.e., every vertex is of degree at most Δ. We try to figure out some bounded maximum degree graph classes, under which the problem can be solved in polynomial time.
Similar content being viewed by others
References
Alekseev, V.E.: On the local restrictions effect on the complexity of finding the graph independence number. In: Combinatorial-Algebraic Methods in Applied Mathematics, Gorkiy University, pp. 3–13. Russian (1983)
Alekseev, V.E.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discrete Appl. Math. 132(1–3), 17–26 (2004)
Brandstädt, A., Hammer, P.L.: A note on α-redundant vertices in graphs. Discrete Appl. Math. 108(3), 301–308 (2001)
Brandstädt, A., Hoàng, C. T.: On clique separators, nearly chordal graphs, and the maximum weight stable set problem. Theor. Comput. Sci. 389(1–2), 295–306 (2007)
Brandstädt, A., Giakoumakis, V.: Addendum to: Maximum weight independent sets in hole- and co-chair-free graphs. Inform. Process. Lett. 115(2), 345–350 (2015)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. J. Comb. Theory B 34(1), 22–39 (1983)
Karp, R.M.: Reducibility Among Combinatorial Problems. Complexity of Computer Computations, pp 85–103. Springer, USA (1972)
Korshunov, A.D.: Coefficient of internal stability. Kibernetika 10(1), 17–28 (1974). (in Ukrainian)
Lê, N.C., Brause, C., Schiermeyer, I.: New sufficient conditions for α-redundant vertices. Discrete Math. 338(10), 1674–1680 (2015)
Lê, N.C., Brause, C., Schiermeyer, I.: The maximum independent set problem in subclasses of si;j;k-free graphs. Electron. Notes Discrete Math. 39, 43–49 (2015)
Lozin, V.V., Rautenbach, D.: On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM J. Discrete Math. 18(1), 195–206 (2004)
Lozin, V.V., Milanič, M.: Maximum independent sets in graphs of low degree. In: SODA ’07 Proc. 18Th Annu. ACM SIAM Symp. Discret., pp. 874–880, ACM Press (2007)
Lozin, V.V., Milanic̆, M.: On the maximum independent set problem in subclasses of planar graphs. J. Graph Algorithms Appl. 14(2), 269–286 (2010)
Lozin, V.V., Monnot, J., Ries, B.: On the maximum independent set problem in subclasses of subcubic graphs. J. Discrete Algorithms 31, 104–112 (2013)
Acknowledgments
This research is supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam, project code: 101.99-2016.20. The manuscript has been finished in the time the first author was in Vietnam Institute for Advanced Studies in Mathematics, Year 2018 and has been revised in the time the first author was in Institute of Mathematics, Vietnam Academy of Science and Technology, Year 2019. The authors would like to thank to all received supports. We also want to express the appreciation to the anonymous referees and the editors for their very useful advices, comments, and corrections.
Funding
The first author receives funding from Vietnamese Institute for Advanced Studies in Mathematics, from 03-04/2018.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Hoang Tuy
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lê, N.C., Tran, T. On the Maximum Independent Set Problem in Graphs of Bounded Maximum Degree. Acta Math Vietnam 45, 463–475 (2020). https://doi.org/10.1007/s40306-020-00368-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40306-020-00368-0