An algorithm for finding homogeneous pairs

Abstract

A homogeneous pair in a graph G = (V, E) is a pair Q₁, Q₂ of disjoint sets of vertices in this graph such that every vertex of V (Q₁ ∪ Q₂) is adjacent either to all vertices of Q₁ or to none of the vertices of Q₁ and is adjacent either to all vertices of Q₂ or to none of the vertices of Q₂. Also $\text{¦Q}{}_1\text{¦ ⩾ 2}$ or $\text{¦Q}{}_2\text{¦⩾ 2}$ and ¦V ($\text{Q}{}_1\text{∪ Q}{}_2\text{)¦ ⩾ 2}$. In this paper we present an O(mn³)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one.