On decision and optimization ( k, l)-graph sandwich problems

Abstract

A graph G is ( k, l) if its vertex set can be partitioned into at most k independent sets and l cliques. The ( k, l)-Graph Sandwich Problem asks, given two graphs G 1=( V, E 1) and G 2=( V, E 2), whether there exists a graph G=( V, E) such that E 1⊆ E⊆ E 2 and G is ( k, l). In this paper, we prove that the ( k, l)-Graph Sandwich Problem is NP-complete for the cases k=1 and l=2; k=2 and l=1; or k= l=2. This completely classifies the complexity of the ( k, l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+ l>2; the problem is polynomial otherwise. We consider the degree Δ constraint subproblem and completely classify the problem as follows: the problem is polynomial, for k⩽2 or Δ⩽3; the problem is NP-complete otherwise. In addition, we propose two optimization versions of graph sandwich problem for a property Π: MAX-Π-GSP and MIN-Π-GSP. We prove that MIN-(2,1)-GSP is a Max-SNP-hard problem, i.e., there is a positive constant ε, such that the existence of an ε-approximative algorithm for MIN-(2,1)-GSP implies P=NP.