Combinatorial optimization problems in the analysis and design of probabilistic networks

Abstract

This paper presents some results regarding the design of reliable networks. The problem under consideration involves networks which are undirected graphs having equal and independent edge failure probabilities. The index of reliability is the probability that the network fails (becomes disconnected). For "small" edge failure probabilities and given p and q there exists a class of p vertex, q edge graphs with the property that any graph in the class has a smaller probability of disconnection than any graph outside of the class. We solve the problem of synthesizing graphs in this class.

The structure of max ?-minm?+1 graphs used in the design of reliable networks

“…First, we consider the case 2e/n ¢ 4. The following observation is a direct min m l/1 (G) Å m l/1 (H) consequence of Lemma 3 and Theorem 3 of [2]:…”

“…In parallel with Lemma 3, a result concerning a (l / 1)-edge disconnecting set being nearly independent is Lemma 3 [2]. If U is a minimum edge-disconnecting set of a graph G with k Å l, then U is either an incidence the following:…”

“…The constructing problem of a other. It can be easily verified that the disconnected probamax l-min m l graph for given natural numbers n and bility P(G, r) of graph G(n, e) can be expressed as e(¢n) was completely solved by Bauer et al in [2]. Therefore, the design problem of reliable networks for…”

“…In this paper, the concept of the max l-min m l graph is given possible values of e and n. For its construction, we refer the reader to [2]. From this reference, we see that generalized to that of the max l-min m i graph, and a characterization of the max l-min m l/1 graph is given.…”

“…In last case, over all max l graphs. In Section 2, we introduce the concept of a nearly independent edge-disconnecting set H 1 (k) has one node with degree 2k / 2 and n 0 1 nodes with degree 2k / 1. and establish Lemma 5 as a base of the paper, which is parallel to Proposition 1 in [2]. Finally, we prove that all max l-min m l/1 graphs are edges added to G in forming H(n, e) is insufficient to raise the minimum degree so that every general Harary max l-min m l for any l ¢ 2 and give a characterization of max l-min m l/1 graphs when l Å 2 in Section 6.…”