Combining decomposition approaches for the Maximum Weight Stable Set problem

Abstract

The Maximum Weight Stable Set Problem (MWS) is a well-known NP-hard problem. A popular way to study MWS is to detect graph classes for which MWS can be solved in polynomial time. In this context some decomposition approaches have been introduced in the literature, in particular decomposition by homogeneous sets and decomposition by clique separators, which had several applications in the literature. Brandstädt and Hoàng (2007) [6] showed how to combine such two decomposition approaches and stated the following result: If the MWS problem can be solved in polynomial time for every subgraph of a graph G which has no homogeneous set and has no clique separators then so can the problem for G. This result, namely Corollary 9 of [6], was used in various papers.Unfortunately, it was stated in a successive paper by Brandstädt and Giakoumakis (2015) [5] that “Actually, [6] does not give a proof for this, and it seems that Corollary 9 of [6] promises too much; later attempts for proving it failed, and thus, it is unproven and its use has to be avoided.”This manuscript introduces a proof of Corollary 9 of [6]. Furthermore a third decomposition approach, namely decomposition by anti-neighborhoods of vertices, is combined together with such two decomposition approaches. Then the various papers in which Corollary 9 of [6] was used would be fixed.