#### Vol. 11, No. 5, 2018

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The $k$-diameter component edge connectivity parameter

### Nathan Shank and Adam Buzzard

Vol. 11 (2018), No. 5, 845–856
##### 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.

##### Keywords
network reliability, connectivity, conditional connectivity, edge failure, graph theory
##### Mathematical Subject Classification 2010
Primary: 05C05, 05C12, 05C90, 94C15