On parallel recognition of cographs

Abstract

In this paper, we modify a known parallel cograph-recognition algorithm proposed by Nikolopoulos and Palios [S.D. Nikolopoulos, L. Palios, Efficient parallel recognition of cographs, Discrete Applied Mathematics 150 (1–3) (2005) 182–215] and provide a new analysis of the algorithm. Given an input graph G with n vertices and m edges, we obtain the following three results based on our analysis: 1. When G is k -regular for a fixed positive integer k , the cograph-recognition problem can be optimally solved in O ( log n ) time using O ( n + m log n ) processors on an EREW PRAM. 2. When G is k -regular for k = O ( log log n ) , the cograph-recognition problem can be solved in O ( log n log log n ) time using O ( n + m log n ) processors on an EREW PRAM. 3. Given a positive integer α = O ( log log n ) , the cograph-recognition problem can be solved in O ( log n log log n ) time using O ( n + m log n ) processors on an EREW PRAM, provided the number of vertices in G with degree larger than α is at most O ( log n ) . The above results improve upon the previously best known result, which took O ( log 2 n ) time using O ( n + m log n ) processors on an EREW PRAM.