Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

Abstract

AbstractThe Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H, $$G+H$$ G + H denotes the disjoint union of G and H. This manuscript shows that (i) WIS can be solved for ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graphs in polynomial time, where a $$P_4$$ P 4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graph G there is a family $${{\mathcal {S}}}$$ S of subsets of V(G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of $${\mathcal {S}}$$ S . These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain $$S_{i,j,k}$$ S i , j , k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.