On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem

Abstract

Clique separators in graphs are a helpful tool used by Tarjan as a divide-and-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Maximum Clique, Graph Coloring and Minimum Fill-in, but few examples of graph classes having clique separators are known. We use this method to solve MWS in polynomial time for two classes where the unweighted Maximum Stable Set (MS) Problem is solvable in polynomial time by augmenting techniques but the complexity of the MWS problem was open. Another example, namely a result by Alekseev for the MWS problem on a subclass of P 5 -free graphs obtained by clique separators, can be improved by our techniques. We also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion: A graph is nearly Π if for each of its vertices, the subgraph induced by the set of its nonneighbors has property Π . We deal with the cases Π ∈ {chordal, perfect}. This also simplifies a result obtained by a method called struction.

On Independent Vertex Sets in Subclasses of Apple-Free Graphs

“…Throughout this section, let G be a prime (A 4 , P 6 )-free atom. Our results in this subsection improve and extend corresponding results in [3,27].…”

“…In [3,7], some examples are given where this technique can be applied. We also use the following notion: Let denote a graph property.…”

“…In [30], k-apples were called k-pans. The 4-apple is also called P or banner in various papers [1,3,4,15,20]. Thus, a graph is apple-free if it contains no induced subgraph A k for any k ≥ 4, i.e., it is (A 4 , A 5 , A 6 , .…”

“…Odd apples were introduced by De Simone in [11], and Olariu in [30] called the apple-free graphs pan-free. There are various papers studying the complexity of the MWS problem for A 4 -free graphs with additional requirements such as being K 2,3 -free and C k -free for all k ≥ 5 [22], C k -free for all k ≥ 5 [14], P 5 -free [4,20], P 6 -free [3], P 7 -free [28] etc. Recently, Lozin and Milanič [21,24] studied apple-free graphs with degree constraints and showed that for every fixed k and , MWS is polynomially solvable for (A k , A k+1 , .…”

“…Whenever MWS is efficiently solvable on the atoms of a graph G, it is efficiently solvable on G. However, few examples are known where this approach could be applied for obtaining a polynomial time MWS algorithm on a graph class. In [3], it is shown that the atoms can additionally be assumed to be prime graphs in the sense of modular decomposition, i.e., whenever MWS is efficiently solvable on the prime atoms of a graph G, it is efficiently solvable on G. More exactly:…”