Efficient parallel algorithms for several intersection graphs

A study is made of several classes of intersection graphs, i.e. interval graphs, circular-arc graphs, and some of their subclasses. It is shown that recognizing a proper interval graph, a unit interval graph, and a claw-free interval graph is in NC. In fact, the recognition can be done in O(log² n) time with O(n³) processors. The corresponding interval representation can be obtained within the same time bound if O(n⁴) processors are used. Some necessary and sufficient conditions for the minimum interval representations are obtained. An NC algorithm is presented which constructs a circular-arc representation for the graph whose augmented adjacency matrix has quasi-circular 1´s or has the consecutive 0´s property