Complexity issues for the sandwich homogeneous set problem

Abstract

Graph sandwich problems were introduced by Golumbic et al. (1994) in [12] for DNA physical mapping problems and can be described as follows. Given a property Π of graphs and two disjoint sets of edges E 1 , E 2 with E 1 ⊆ E 2 on a vertex set V , the problem is to find a graph G on V with edge set E s having property Π and such that E 1 ⊆ E s ⊆ E 2 . In this paper, we exhibit a quasi-linear reduction between the problem of finding an independent set of size k ≥ 2 in a graph and the problem of finding a sandwich homogeneous set of the same size k . Using this reduction, we prove that a number of natural (decision and counting) problems related to sandwich homogeneous sets are hard in general. We then exploit a little further the reduction and show that finding efficient algorithms to compute small sandwich homogeneous sets would imply substantial improvement for computing triangles in graphs.