The $k$-diameter component edge connectivity parameter

Abstract

We focus on a network reliability measure based on edge failures and considering a network operational if there exists a component with diameter $k$ or larger. The $k$-diameter component edge connectivity parameter of a graph is the minimum number of edge failures needed so that no component has diameter $k$ or larger. This implies each resulting vertex must not have a $k$-neighbor. We give results for specific graph classes including path graphs, complete graphs, complete bipartite graphs, and a surprising result for perfect $r$-ary trees.