The edge fault-diameter of Cartesian graph bundles

Abstract

A Cartesian graph bundle is a generalization of a graph covering and a Cartesian graph product. Let $G$ be a $k_G$-edge connected graph and ${\bar{\mathcal{D}}}_c\left(G\right)$ be the largest diameter of subgraphs of $G$ obtained by deleting $c<k_G$ edges. We prove that ${\bar{\mathcal{D}}}_{a+b+1}\left(G\right)\le {\bar{\mathcal{D}}}_a\left(F\right)+{\bar{\mathcal{D}}}_b\left(B\right)+1$ if $G$ is a graph bundle with fibre $F$ over base $B$, $a<k_F$, and $b<k_B$. As an auxiliary result we prove that the edge-connectivity of graph bundle $G$ is at least $k_F+k_B$.