The Pair Completion algorithm for the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e. a nonempty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G1(V, E1) and G2(V,E2), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V, ES), E1 ⊆ ES ⊆ E2, which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [14] to solve the problem in O(n min{|E1|, |E2|} log n) time, thus outperforming the other known HSSP deterministic algorithms for inputs where max{|E1|, |E2|} = Ω(n log n).