Partitions and well-coveredness: The graph sandwich problem

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