A Fast Parallel Algorithm to Recognize P4-sparse Graphs

A number of problems in computational semantics, group-based collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain \local density" characteristics. Typically, the notion of local density is equated with the absence of chordless paths of length three or more. Recently, a new metric for local density has been proposed, allowing a number of such induced paths to occur. More precisely, a graph G is called P4-sparse if no set of ve vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize the well-known class of cographs corresponding to a more stringent local density metric. One remarkable feature of P4-sparse graphs is that they admit a tree representation unique up to isomorphism. In this work we present a parallel algorithm to recognize P4-sparse graphs and show how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation. With a graph G = (V; E) with jV j = n and jE j = m as input, our algorithms run in O(log n) time using O n 2 +mn log n processors in the EREW-PRAM model.