Efficient parallel recognition of cographs

In this paper, we establish structural properties for the class of complement reducible graphs or cographs, which enable us to describe efficient parallel algorithms for recognizing cographs and for constructing the cotree of a graph if it is a cograph; if the input graph is not a cograph, both algorithms return an induced P4. For a graph on n vertices and m edges, both our cograph recognition and cotree construction algorithms run in O(log2n) time and require O((n+m)/logn) processors on the EREW PRAM model of computation. Our algorithms are motivated by the work of Dahlhaus (Discrete Appl. Math. 57 (1995) 29-44) and take advantage of the optimal O(logn)-time computation of the co-connected components of a general graph (Theory Comput. Systems 37 (2004) 527-546) and of an optimal O(logn)-time parallel algorithm for computing the connected components of a cograph, which we present. Our results improve upon the previously known linear-processor parallel algorithms for the problems (Discrete Appl. Math. 57 (1995) 29-44; J. Algorithms 15 (1993) 284-313): we achieve a better time-processor product using a weaker model of computation and we provide a certificate (an induced P4) whenever our algorithms decide that the input graphs are not cographs.