Independent sets in (P6, diamond)-free graphs

An independent set (or a stable set) in a graph G is a subset of pairwise nonadjacent vertices of G. An independent set of G is maximal if it is not properly contained in any other independent set of G. The Maximum-Weight Independent Set (WIS) problem is the following: Given a graphG = (V,E) and a weight function w on V , determine an independent set of G of maximum weight. Let αw(G) denote the maximum weight of an independent set of G. The WIS problem reduces to the IS problem if all vertices v have the same weight w(v) = 1. The WIS problem is NP-hard [34] and remains difficult for cubic graphs [27] and for planar graphs [26], while it can be efficiently solved for various graph classes which include perfect graphs [33] (and the class of perfect graphs includes the chordal graphs),K1,3-free graphs [2, 37, 40, 42, 45], and 2K2-free graphs [21, 22, 38]. The Minimum-Weight Independent Dominating Set (WID) problem is the following: Given a graph G = (V,E) and a weight function w on V , determine a maximal independent set of G of minimum weight. Let ιw(G) denote the minimum weight of a maximal independent set of G. The WID problem reduces to the ID problem if all vertices v have the same weight w(v) = 1. The WID problem is NP-hard [28] and remains difficult for chordal graphs [18] and for 2P3-free perfect graphs [46], while it can be efficiently solved for various graph classes which include permutation graphs [15], locally independent well-dominated graphs [47], and 2K2-free graphs [21, 22, 38]. Both WIS and WID remain difficult for triangle-free graphs [43]. Also, for both IS and ID, the class of P5-free graphs is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity is an open question (see [1, 3, 7]). On the other hand, several papers introduced structural properties on graphs containing no long induced paths (see e.g. [5, 6, 19, 39]), often applied to design efficient algorithms for solving various NP-hard