Three remarks on W2 graphs

Abstract

Let k≥1. A graph G is Wk if for any k pairwise disjoint independent vertex subsets A1,…,Ak in G, there exist k pairwise disjoint maximum independent sets S1,…,Sk in G such that Ai⊆Si for i∈[k]. Recognizing W1 graphs is coNP-hard, as shown by Chvátal and Slater (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing Wk graphs is coNP-hard for k≥2. On the positive side, we show that recognizing Wk graphs is, for each k≥2, FPT parameterized by clique-width and by tree-width. Finally, we construct graphs G that are not W2 such that, for every vertex v in G and every maximal independent set S in G−N[v], the largest independent set in N(v)∖S consists of a single vertex, thereby refuting a conjecture of Levit and Tankus.