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.

Consider a simple undirected graph G = (V,E) with vertex set V and edge set E. Let G-u be a subgraph induced by the vertex set V-{u}. The distance δG(x,y) is defined as the length of the shortest path between vertices x and y in G. The vertex u ∈ V is a hinge vertex if there exist two vertices x,y ∈ V-{u} such that δG-u(x,y)>δG(x,y). Let U be a set consisting of all hinge vertices of G. The neighborhood of u is the set of all vertices adjacent to u and is denoted by N(u). We define d(u) = max{δG-u(x,y) | δG-u(x,y)>δG(x,y),x,y ∈ N(u)} for u ∈ U as detour degree of u. A maximum detour hinge vertex problem is to find a hinge vertex u with maximum d(u) in G. In this paper, we proposed an algorithm to find the maximum detour hinge vertex on an interval graph that runs in O(n2) time, where n is the number of vertices in the graph.

We investigate connected proper interval graphs without vertex labels. We first give the number of connected proper interval graphs of n vertices. Using this result, a simple algorithm that generates a connected proper interval graph uniformly at random up to isomorphism is presented. Finally an enumeration algorithm of connected proper interval graphs is proposed. The algorithm is based on reverse search, and it outputs each connected proper interval graph in (O)1 time.

A graph is an interval graph if and only if each vertex in the graph can be associated with an interval on the real line such that any two vertices are adjacent in the graph exactly when the corresponding intervals have a nonempty intersection. A number of interesting applications for interval graphs have been found in the literature. In order to find structural features common to structural data which can be represented by intervals, this paper proposes new interval graph structured patterns, called linear interval graph patterns, and a polynomial time algorithm for finding a minimally generalized linear interval graph pattern explaining a given finite set of interval graphs.

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 can be used to identify critical nodes in an actual network. A number of studies concerning hinge vertices have been made in recent years. In general, it is known that more efficient sequential or parallel algorithms can be developed by restricting classes of graphs. For instance, Chang et al. presented an O(n+m) time algorithm for finding all hinge vertices of a strongly chordal graph. Ho et al. presented a linear time algorithm for all hinge vertices of a permutation graph. In this paper, we shall propose a parallel algorithm which runs in O(log n) time with O(n) processors on CREW PRAM for finding all hinge vertices of an interval graph.