A Full Characterization of Irrelevant Components in Diameter Constrained Reliability

In classical network reliability analysis, the system under study is a network with perfect nodes but imperfect link, that fail stochastically and independently. There, the goal is to find the probability that the resulting random graph is connected, called \emph{reliability}. Although the exact reliability computation belongs to the class of $\mathcal{NP}$-Hard problems, the literature offers three exact methods for exact reliability computation, to know, Sum of Disjoint Products (SDPs), Inclusion-Exclusion and Factorization. Inspired in delay-sensitive applications in telecommunications, H\'ector Cancela and Louis Petingi defined in 2001 the diameter-constrained reliability, where terminals are required to be connected by $d$ hops or less, being $d$ a positive integer, called diameter. Factorization theory in classical network reliability is a mature area. However, an extension to the diameter-constrained context requires at least the recognition of irrelevant links, and an extension of deletion-contraction formula. In this paper, we fully characterize the determination of irrelevant links. Diameter-constrained reliability invariants are presented, which, together with the recognition of irrelevant links, represent the building-blocks for a new factorization theory. The paper is closed with a discussion of trends for future work.