Vertex and Mixed $k$-Diameter Component Connectivity

TL;DR

This paper introduces and analyzes the vertex and mixed $k$-diameter component connectivity parameters, determining the minimum failures needed to reduce network diameter below a threshold for various graph classes.

Contribution

It defines the vertex and mixed variants of the $k$-diameter component connectivity and computes these parameters for specific graph classes.

Findings

Vertex $k$-diameter component connectivity values for paths, cycles, complete, and bipartite graphs.

Mixed variant results for the same graph classes.

Analysis of perfect $r$-ary trees for the vertex variant.

Abstract

n the $k$-diameter component connectivity model a network is consider operational if there is a component with diameter at least $k$. Therefore, a network is in a failure state if every component has diameter less than $k$. In this paper we find the vertex variant of the $k$-diameter component connectivity parameter, which is the minimum number of vertex deletions in order to put a network into a failure state, for particular classes of graphs. We also show the mixed variant by allowing vertex and edge failures within the network. We show results for paths, cycles, complete, and complete bipartite graphs for both variants as well as perfect $r$-ary trees for the vertex variant.