Diameter-related properties of graphs and applications to network reliability theory LOUIS PETINGI College of Staten Island City University of New York Computer Science Department 2800 Victory Boulvard, Staten Island, N.Y. USA louis.petingi@csi.cuny.edu Abstract: Given an undirected graph G =(V,E), two distinguished vertices s and t of G, and a diameter bound D,a D-s, t-path is a path between s and t composed of at most D edges. An edge e is called D-irrelevant if does not belong to any D-s, t-path of G. In this paper we study the problem of efficiently detecting D-irrelevant edges and also study the computational complexity of diameter-related problems in graphs. Detection and subsequent deletion of D-irrelevant edges have been shown to be fundamental in reducing the computational effort to evaluate the Source-to-terminal Diameter-Constrained reliability of a graph G, R {s,t} (G, D), which is defined as the probability that at least a path between s and t, with at most D edges, survives after deletion of the failed edges (under the assumption that edges fail independently and nodes are perfectly reliable). Among other results, we present sufficient conditions to efficiently recognize irrelevant edges and we present computational results illustrating the importance of embedding a procedure to detect irrelevant edges based on these conditions, within the frame of an algorithm to calculate R {s,t} (G, D), built on a theorem of Moskowitz. These results yield a research path for the theoretical study of the problem of determining families of topologies in which R {s,t} (G, D) can be computed in polynomial time, as the general problem of evaluating this reliability measure is NP-Hard. Key–Words: Network reliability, diameter constraint, paths, factoring, topological reductions. 1 Introduction Unless otherwise stated, in this paper we consider undirected graphs G =(V,E), where V represents a finite set of vertices and E is a finite set of edges. The purpose of this work is two-fold. We first investigate, from a computational point of view, di- ameter properties of graphs related to the following optimization problem: given two vertices s and t of G, we would like to efficiently identify edges that do not belong to any path between s and t of length less or equal to a given bound D; we then apply some of the results shown to compute more efficiently the Diameter-Constrained reliability (DCR) of a com- munication network (originally introduced in [17]), a constrained version of the classical network reliabil- ity measure (refer to [20, 21, 22, 24, 25] for further discussion on this classical model). This study serves as a guide to address the problem of identifying fami- lies of topologies in which the DCR can be computed in polynomial time (as its computation is known to be NP-Hard). Given a probabilistic graph G =(V,E), a set of terminal nodes K ⊆ V , and a diameter bound D, in which each edge e ∈ E has been assigned a probability of failure q e =1 − r e (r e is called the reliability of the edge e) under the assumption that edges fail independently and nodes are perfectly re- liable, the K-terminal Diameter-Constrained reliabil- ity, R K (G, D), gives the probability of the event that for each pair of nodes x, y ∈ K, a path between x and y of length (i.e., number of edges comprising the path) D or less, called a D-x, y-path, survives after deletion of the failed edges. In this paper we consider the case when K = {s, t}, known as the Source-to- terminal Diameter-Constrained reliability of a graph G, denoted as R {s,t} (G, D). For the classical relia- bility measure, the K-terminal reliability R K (G), of a graph G, is the probability that after the removal of the failed edges, each pair of nodes x, y ∈ K is connected by at least an operational path, indepen- dently of its length. Both the classical reliability and the DCR can be computed by application of a theo- rem of Moskowitz [14], also refered as the Factoring Theorem, in which the reliability of the probabilistic WSEAS TRANSACTIONS on MATHEMATICS Louis Petingi E-ISSN: 2224-2880 884 Issue 9, Volume 12, September 2013