Polynomial algorithms for the maximum stable set problem on particular classes of P5-free graphs

doi:10.1016/S0020-0190(96)00197-4

Information Processing Letters

Volume 61, Issue 3, 14 February 1997, Pages 137-143

https://doi.org/10.1016/S0020-0190(96)00197-4 Get rights and content

Abstract

The Maximum Stable Set Problem (MS) is a well-known NP-hard problem. A popular research stream considers classes of graphs, defined in terms of forbidden subgraphs, in which either MS is NP-hard or can be solved by polynomial algorithms. In this paper we focus on three of these classes: in one of them, MS is still NP-hard; in the others, the problem difficulty is an open question. However, for those graphs in such classes that have no induced paths with five vertices (P₅) the problem becomes solvable in polynomial time. Whether MS is difficult or not in the class of P₅-free graphs is still open.

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Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem; our results improve and extend various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are $P_{5}$ -free (the $P_{5}$ is the induced path with five vertices) and obtain new polynomial time results for the MWS problem. The complexity of this problem on the class of $P_{5}$ -free graphs is still unknown.

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^☆: This work has been partially supported by CNR Progetto Finalizzato Trasporti 2, Contract N. 93.01781.PFZ4.

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Polynomial algorithms for the maximum stable set problem on particular classes of P5-free graphs☆

Abstract

Discrete Appl. Math.

Theoret. Comput. Sci.

Ann. Discrete Math.

J. Combin. Theory Ser. B

Polynomial algorithms for special cases of the balanced complete bipartite subgraph problem

Univ. di L'Aquila, Dipartimento di Matematica Pura e Applicata, Tech. Rept. n. 58

The maximum stable set problem for particular graphs with no induced P5

Dominating cliques in P5-free graphs

Period. Math. Hungar.

The complexity of generalized clique packing

Discrete Appl. Math.

Polynomial algorithms for the maximum stable set problem on particular classes of P₅-free graphs☆

The maximum stable set problem for particular graphs with no induced P₅

Dominating cliques in P₅-free graphs