Diameter bounds for altered graphs

Abstract

AbstractThe main question addressed in this article is the following: If t edges are removed from a (t + 1) edge‐connected graph G having diameter D, how large can the diameter of the resulting graph be? (The diameter of a graph is the maximum, over all pairs of vertices, of the length of the shortest path joining those vertices.) We provide bounds on this value that imply that the maximum possible diameter of the resulting graph, for large D and fixed t, is essentially (t + 1) · D. The bulk of the proof consists of showing that, if t edges are added to an n‐vertex path Pn, then the diameter of the resulting graph is at least (n/(t + 1)) ‐ 1. Using a similar proof, we also show that if t edges are added to an n‐vertex cycle Cn, then the least possible diameter of the resulting graph is (for large n) essentially n/(t + 2) when t is even and n/(t + 1) when t is odd. Examples are given in all these cases to show that there exist graphs for which the bounds are achieved. We also give results for the corresponding vertex deletion problem for general graphs. Such results are of interest, for example, when studying the potential effects of node or link failures on the performance of a communication network, especially for networks in which the maximum time‐delay or signal degradation is directly related to the diameter of the network.