Maximum weight independent set for ℓclaw-free graphs in polynomial time

The MaximumWeight Independent Set (MWIS) problem is a well-known NPhard problem. A popular way to study MWIS is to detect graph classes for which MWIS 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. For graphs G1, G2, G1 + G2 denotes the disjoint union of G1 and G2, and for a constant l ≥ 2, lG denotes the disjoint union of l copies of G. A claw has vertices a, b, c, d, and edges ab, ac, ad. MWIS can be solved for claw-free graphs in polynomial time; the first two polynomial time algorithms were introduced in 1980 by [22, 29], then revisited by [24], and recently improved by [9, 10], and by [25, 26] with the best known time bound in [26]. Furthermore MWIS can be solved for the following extensions of claw-free graphs in polynomial time: fork-free graphs [19], K2+claw-free graphs [20], and apple-free graphs [6, 7]. This manuscript shows that for any constant l, MWIS can be solved for lclawfree graphs in polynomial time. Our approach is based on Farber’s approach showing that every 2K2-free graph has O(n ) maximal independent sets [11], which directly leads to a polynomial time algorithm for MWIS on 2K2-free graphs by dynamic programming. Solving MWIS for lclaw-free graphs in polynomial time extends known results for claw-free graphs, for lK2-free graphs for any constant l [2, 12, 28, 30], for K2+claw-free graphs, for 2P3-free graphs [21], and solves the open questions for 2K2+P3-free graphs and for P3+claw-free graphs being two of the minimal graph classes, defined by forbidding one induced subgraph, for which the complexity of MWIS was an open problem.