Maximizing k-Terminal Network Reliability in Some Sparse Graphs

k -terminal network reliability is the probability that k terminal vertices are connected given that edges in the network fail independently while vertices do not fail. It depends on the distribution of these terminal vertices as well as network topology. The problem of computing k -terminal reliability is NP-hard. Previous literature mainly focus on designing efficient algorithms to compute it in different graphs, but are lacking in the analysis for optimal distribution of k terminal vertices in sparse graphs, within which those with n nodes and m edges, where m \leq n + 1 , are most basic classes. Hence, it is of great significance to investigate the optimal distribution of terminal vertices in these classes of graphs before considering general cases. In this paper, we prove that k -terminal network reliability obtains the maximum if k terminal vertices induce a connected subgraph. Further, we give equations of maximum reliability for all possible graphs in the above classes. The experiments illlustrate the variation, with several parameters and according to our theoretical results, of maximal k -terminal reliability, and provide observations for graphs with any number of edges.