Partitions and well-coveredness: The graph sandwich problem

Abstract

A graph G is well-covered if every maximal independent set of G is maximum. A (k,ℓ)-partition of a graph G is a partition of its vertex set into k independent sets and ℓ cliques. A graph is (k,ℓ)-well-covered if it is well-covered and admits a (k,ℓ)-partition. The recognition of (k,ℓ)-well-covered graphs is polynomial-time solvable for the cases (0,1), (0,2), (1,0), (1,1), (1,2), and (2,0), and hard, otherwise. In the graph sandwich problem for a property Π, we are given a pair of graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2, and asked whether there is a graph G=(V,E) with E1⊆E⊆E2, such that G satisfies the property Π. The problem of recognizing whether a graph G satisfies a property Π is equivalent to the particular graph sandwich problem where E1=E2. In this paper, we study the graph sandwich problem for the property of being (k,ℓ)-well-covered. We present some structural characterizations and extending previous studies on the recognition of (k,ℓ)-well-covered graphs, we prove that Graph Sandwich for(k,ℓ)-well-coveredness is polynomial-time solvable when (k,ℓ)∈{(0,1),(1,0),(1,1),(0,2)}. Besides, we show that it is NP-complete for the property of being (1,2)-well-covered.