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In network reliability analysis, an important problem is to determine the probability that a specified subset of vertices in an undirected graph is connected. It is well known that, by using Moskowitz's factoring theorem, the reliability of a graph can be expressed in terms of the reliabilities of a graph with one fewer vertex and another with one fewer edge. The theorem can be applied recursively on the reduced graphs. The computations involved in this recursion can be represented by a binary structure such that its leaves correspond to reduced graphs whose reliabilities can be readily evaluated. In general, as the recursion progresses, series and parallel edges are created which can be reduced by using series and parallel rules of reliability assuming edges fail independently of each other. The computational complexity is a function of the number of leaves in the binary structure, and for a given graph, an optimal binary structure is the one with minimal number of leaves. In this article, a combinatorial invariant of a graph, called the domination, is established. Several important properties of the domination with regard t o the topology of the graph are investigated. It is shown that for a given graph, the number of leaves in the optimal binary structure is equal t o the domination of the graph and recursive application of the factoring theorem yields an optimal structure if and only if at each step the reduced graphs generated have nonzero dominations. Finally, an algorithm is presented that guarantees optimal binary structure generation and therefore an efficient implementation of the factoring theorem to compute the network reliability. PRELIMINARIESConsider an undirected graph G = (V, E) with vertex set V = {ul, u2, . . . , u,} and edge set E = {el, e,, . . . , eb}. Vertices do not fail, but at an instant of interest, an edge ei has reliability pi, independent of the states of the other edges. Let K be a specified subset of V with IKI 2 2. The K-terminal reliabiliw of G, denoted by RK(G), is the probability that the vertices in K are connected. A success set is a minimal set of edges of G such that the vertices in K are connected; the set is minimal in the sense that deletion of any edge causes the vertices in K to be disconnected. A failure set is defined analogously.Topologically, a success set is a minimal tree of C covering all vertices in K. We call such a tree a K-tree, to distinguish it from other trees of C . Thus a K-tree is a tree of

The purpose of this article is to introduce several results concerning the analysis and synthesis of reliable or invulnerable networks. First, the notion of signed reliability domination of systems is described and some applications to reliability analysis are reviewed. Then the analysis problem is considered and a brief summary of the difficulty of calculating various reliability measures is presented. Some relevant concepts in the synthesis of a most reliable network are studied. The article concludes with an introduction to a non-probabilistic approach to evaluate the vulnerability of a network.

Abstract. Let G (V, E) be a graph whose edges may fail with known probabilities and let K _ V be specified. The K-terminal reliability of G, denoted R(GK), is the probability that all vertices in K are connected. Computing R(G:) is, in general, NP-hard. For some series-parallel graphs, R(Gn) can be computed in polynomial time by repeated application of well-known reliability-preserving reductions.However, for other series-parallel graphs, depending on the configuration of K, R (Gn) cannot be computed in this way. Only exponential-time algorithms as used on general graphs were known for computing R (G<) for these "irreducible" series-parallel graphs. We prove that R(Gn) is computable in polynomial time in the irreducible case, too. A new set of reliability-preserving "polygon-to-chain" reductions of general applicability is introduced which decreases the size of a graph, and conditions are given for a graph admitting such reductions. Combining all types of reductions, an O(IEI) algorithm is presented for computing the reliability of any series-parallel graph irrespective of the vertices in K.

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