Partitions and Well-coveredness: The Graph Sandwich Problem

Sancrey Alves , Fernanda Couto , Luerbio Faria , Sylvain Gravier (1, 2, 3) , Sulamita Klein , Uéverton Souza , Rodrigo Marinho

1 CNRS and Université Grenoble Alpes, 38041 Grenoble, France
2 IF - Institut Fourier
3 SFR MAM - Maths à Modeler

A graph is well-covered if every maximal independent set is also 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)-wellcovered 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 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 par- ticular 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 Graph Sandwich is NP-complete for the property of being (1, 2)-well-covered.

Domaines: Mathématique discrète [cs.DM]

Sancrey Alves, Fernanda Couto, Luerbio Faria, Sylvain Gravier, Sulamita Klein, et al.. Partitions and Well-coveredness: The Graph Sandwich Problem. Discrete Mathematics, 2023, 346 (3), ⟨10.1016/j.disc.2022.113253⟩. ⟨hal-03667189⟩