The Pair Completion algorithm for the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G 1 ( V , E 1 ) and G 2 ( V , E 2 ) , the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph G S ( V , E S ) , E 1 ⊆ E S ⊆ E 2 , which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17–22] to solve the problem in O ( n min { | E 1 | , | E ¯ 2 | } log n ) time, thus outperforming the other known HSSP deterministic algorithms for inputs where max { | E 1 | , | E ¯ 2 | } = Ω ( n log n ) .