Note on the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-unitary, proper subset H of a graph's vertices such that all vertices in H have the same neighbors outside H. Given two graphs G1(V,E1), G2(V,E2), the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph GS(V,ES), E1⊆ES⊆E2, which has a homogeneous set. Recently, Tang et al. [Inform. Process. Lett. 77 (2001) 17-22] proposed an interesting O(▵1⋅n2) algorithm for this problem, which has been considered its most efficient algorithm since. We show the incorrectness of their algorithm by presenting three counterexamples.