On parallel recognition of cographs

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(logn) time using O(n+mlogn) processors on an EREW PRAM. 2.When G is k-regular for k=O(loglogn), the cograph-recognition problem can be solved in O(lognloglogn) time using O(n+mlogn) processors on an EREW PRAM. 3.Given a positive integer @a=O(loglogn), the cograph-recognition problem can be solved in O(lognloglogn) time using O(n+mlogn) processors on an EREW PRAM, provided the number of vertices in G with degree larger than @a is at most O(logn). The above results improve upon the previously best known result, which took O(log^2n) time using O(n+mlogn) processors on an EREW PRAM.

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