Let G be a probabilistic graph, in which the vertices fail independently with known probabilities. Let K represent a specified subset of vertices. The K-terminal reliability of G is defined as the probability that all vertices in K are connected. When |K|=2, the K-terminal reliability is called the 2-terminal reliability, which is the probability that the source vertex is connected to the destination vertex. The problems of computing K-terminal reliability and 2-terminal reliability have been proven to be #P-complete in general. This work demonstrates that on multi-tolerance graphs, the 2-terminal reliability problem can be solved in polynomial-time and the results can be extended to the K-terminal reliability problem on bounded multi-tolerance graphs.

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.

Given a simple connected graph G with n vertices, the spanning tree problem involves finding a tree that connects all the vertices of G. Solutions to this problem have applications in electrical power provision, computer network design, circuit analysis, among others. It is known that highly efficient sequential or parallel algorithms can be developed by restricting classes of graphs. Circular trapezoid graphs are proper superclasses of trapezoid graphs. In this paper, we propose an O(n) time algorithm for the spanning tree problem on a circular trapezoid graph. Moreover, this algorithm can be implemented in O(log n) time with O(n/log n) processors on EREW PRAM computation model.

In an undirected graph, the feedback vertex set (FVS for short) problem is to find a set of vertices of minimum cardinality whose removal makes the graph acyclic. The FVS has applications to several areas such that combinatorial circuit design, synchronous systems, computer systems, VLSI circuits and so on. The FVS problem is known to be NP-hard on general graphs but interesting polynomial solutions have been found for some special classes of graphs. In this paper, we present an O(n2.68 + γn) time algorithm for solving the FVS problem on trapezoid graphs, where γ is the total number of factors included in all maximal cliques.

Given a simple graph G with n vertices, m edges and k connected components. The spanning forest problem is to find a spanning tree for each connected component of G. This problem has applications to the electrical power demand problem, computer network design, circuit analysis, etc. An optimal parallel algorithm for finding a spanning tree on the trapezoid graph is given by Bera et al., it takes O(log n) time with O(n/log n) processors on the EREW (Exclusive-Read Exclusive-Write) PRAM. Bera et al.'s algorithm is very simple and elegant. Moreover, it can correctly construct a spanning tree when the graph is connected. However, their algorithm can not accept a disconnected graph as an input. Applying their algorithm to a disconnected graph, Concurrent-Write occurs once for each connected component, thus this can not be achieved on EREW PRAM. In this paper we present an O(log n) time parallel algorithm with O(n/log n) processors for constructing a spanning forest on trapezoid graph G on EREW PRAM even if G is a disconnected graph.

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) processors on CREW PRAM for finding all hinge vertices of a trapezoid graph.

This paper analyzes what structural features of graph problems allow efficient parallel algorithms. We survey some parallel algorithms for typical problems on three kinds of graphs, outerplanar graphs, trapezoid graphs and in-tournament graphs. Our results on the shortest path problem, the longest path problem and the maximum flow problem on outerplanar graphs, the minimum-weight connected dominating set problem and the coloring problem on trapezoid graphs and Hamiltonian path and Hamiltonian cycle problem on in-tournament graphs are adopted as working examples.