Let G be a graph and K be a set of target vertices of G. Assume that all vertices of G, except the vertices in K, may fail with given probabilities. The K-terminal reliability of G is the probability that all vertices in K are mutually connected. This reliability problem is known to be #P-complete for general graphs. This work develops the first polynomial-time algorithm for computing the K-terminal reliability of circular-arc graphs.

Let G =(V, E) be an undirected simple graph with u ∈ V. If there exist any two vertices in G whose distance becomes longer when a vertex u is removed, then u is defined as a hinge vertex. Finding the set of hinge vertices in a graph is useful for identifying critical nodes in an actual network. A number of studies concerning hinge vertices have been made in recent years. In a number of graph problems, it is known that more efficient sequential or parallel algorithms can be developed by restricting classes of graphs. In this paper, we shall propose a parallel algorithm which runs in O(log n) time with O(n/log n) processors on EREW PRAM for finding all hinge vertices of a circular-arc graph.