New Results on Independent Sets in Extensions of 2K_2-free Graphs

Abstract

The 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 is solvable 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. For any two graphs G and H, G+H denotes the disjoint union of G and H. A lK_2 is the disjoint union of l edges. A Y_{m,m} is the disjoint union of two stars of m+1 vertices plus one vertex that is adjacent only to the centers of such stars. For any graph family {{{mathcal {Y}}}}, the class of {mathcal Y}-free graphs is formed by graphs which are Y-free for every Y in {{{mathcal {Y}}}}, and the class of lK_2+{{{mathcal {Y}}}}-free graphs is formed by graph which are lK_2+Y-free for every Y in {mathcal Y}. The main result of this manuscript is the following: For any constant m and for any graph family {{{mathcal {Y}}}} which contains an induced subgraph of Y_{m,m}, if WIS is solvable in polynomial time for {{{mathcal {Y}}}}-free graphs, then WIS is solvable in polynomial time for lK_2+{{{mathcal {Y}}}}-free graphs for any constant l. That extends some known polynomial results, namely, when {{{mathcal {Y}}}} = {Y} and Y is a fork or is a P_5. The proof of the main result is based on Farber’s approach to prove that every 2K_2-free graph has O(n^2) maximal independent sets (Farber in Discrete Math 73:249–260, 1989), which directly leads to a polynomial time algorithm to solve WIS for 2K_2-free graphs through a dynamic programming approach, and on some extensions of Farber’s approach.