On (k,ℓ)-Graph Sandwich Problems

Abstract

In this work we consider the Golumbic, Kaplan and Shamir graph sandwich decision problem for property Π, where given two graphs G ¹ = (V,E ¹) and G ² = (V,E ²), the question is to know whether there exists a graph G = (V,E) such that E ¹ ⊆ E ⊆ E ² and G satisfies property Π. The main property Π we are interested in this paper is “being a (k,ℓ)-graph”. We say that a graph G = (V,E) is (k,ℓ) if there is a partition of the vertex set of G into at most k independent sets and at most ℓ cliques. We prove that the strongly chordal-(2,ℓ) graph sandwich problem is NP-complete, for ℓ ≥ 1, and that the chordal-(k,ℓ) graph sandwich problem is NP-complete, for k ≥ 2   , ℓ ≥ 1. We also introduce in this paper a new work proposal related to graph sandwich problems: the graph sandwich problem with boundary conditions. Our goal is to redefine well-known NP-complete graph sandwich problems by cleverly assigning properties to its input graphs so that the redefined problems are polynomially solvable. Let poly-color(k) denote an infinite family of graphs G for which deciding whether G is k-colorable can be done in O(p(n)) time, where p is a polynomial and n = |V(G)|. In order to illustrate how boundary conditions can change the complexity status of a graph sandwich problem, we present here a polynomial-time solution for the (k,ℓ)-graph sandwich problem for all k,ℓ, when beforehand we know that G ¹ belongs to poly-color(k) and G ² has a polynomial number maximal of cliques.