Edge-packing of graphs and network reliability

Charles J. Colbourn
<span title="">1988</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The reliability of a network can be efficiently bounded using graph-theoretical techniques based on edge-packing. We examine the application of combinatorial theorems on edgepacking spanning trees, s, t-paths, and s, t-cuts to the determination of reliability bounds. The application of spanning trees has been studied by Polesskii, and the application of s, t-paths has been studied by Brecht and Colbourn. The use of edge-packings of s, t-cutsets has not been previously examined. We compare the
more &raquo; ... sulting bounds with known bounds produced by different techniques, and establish that the edge-packing bounds often produce a substantial improvement. We also establish that three other edge-packing problems arising in reliability bounding are NP-complete, namely edge-packing by network cutsets, Steiner trees, and Steiner cutsets. Edge-packings of graphs An edge-packing of a multigraph G is a collection of edge-disjoint subgraphs of G. Many combinatorial problems can be viewed as edge-packing, including the chromatic index problem and the independent set problem. In this paper, we are concerned with edge-packing problems which arise in problems concerning network reliablity. The three problems of most interest to us here are edgepackings by spanning trees, by s, t-paths, and by s, t-cuts. We first review the use of edge-packings in the reliability context. A probabilistic graph is a multigraph together with a probability of operation for each edge. Vertices of the multigraph represent communication centers which never fail, while edges represent undirected communication links which operate statistically independently with the stated probabilities. An all-terminal operation in such a graph requires that all vertices be able to communicate with one another via paths of operational edges, and all-terminal reliability is the probability that the network supports an all-terminal operation. A 2-terminal operation for specified vertices s and t requires that there be an operational s, t-path, and 2-terminal reliability is the probability that a 2-terminal operation for s and t can be performed. A k-terminal operation for a specified set of k target nodes requires that each pair of target nodes be able to communicate, and k-terminal reliability is the probability of being able to carry out a k-terminal opeation. The evaluation of
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