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:

• When $G$ is $k$-regular for a fixed positive integer $k$, the cograph-recognition problem can be optimally solved in $O(logn)$ time using $O\left(\frac{n+m}{logn}\right)$ processors on an EREW PRAM.

• When $G$ is $k$-regular for $k=O(loglogn)$, the cograph-recognition problem can be solved in $O(lognloglogn)$ time using $O\left(\frac{n+m}{logn}\right)$ processors on an EREW PRAM.

• Given a positive integer $\alpha =O(loglogn)$, the cograph-recognition problem can be solved in $O(lognloglogn)$ time using $O\left(\frac{n+m}{logn}\right)$ processors on an EREW PRAM, provided the number of vertices in $G$ with degree larger than $\alpha$ is at most $O(logn)$.

The above results improve upon the previously best known result, which took $O\left({log}^{2}n\right)$ time using $O\left(\frac{n+m}{logn}\right)$ processors on an EREW PRAM.