Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

TL;DR

This paper establishes improved upper bounds for the fault diameters of Cartesian graph bundles, relating the fault diameters of the bundle to those of its fiber and base graphs under certain connectivity conditions.

Contribution

It provides new bounds for vertex and edge fault diameters of Cartesian graph bundles, extending previous results with sharper inequalities under connectivity assumptions.

Findings

Upper bounds for vertex fault diameter of Cartesian bundles derived

Upper bounds for edge fault diameter of Cartesian bundles derived

Results depend on connectivity and fault diameter conditions of fiber and base graphs

Abstract

Mixed fault diameter of a graph $G$, $ \D_{(a,b)}(G)$, is the maximal diameter of $G$ after deletion of any $a$ vertices and any $b$ edges. Special cases are the (vertex) fault diameter $\D^V_{a} = \D_{(a,0)}$ and the edge fault diameter $\D^E_{a} = \D_{(0,a)}$. Let $G$ be a Cartesian graph bundle with fibre $F$ over the base graph $B$. We show that (1) $\D^V_{a+b+1}(G)\leq \D^V_{a}(F)+\D^V_{b}(B)$ when the graphs $F$ and $B$ are $k_F$-connected and $k_B$-connected, $0< a < k_F$, $0< b < k_B$, and provided that $\D_{(a-1,1)}(F)\leq \D^{V}_{a} (F)$ and $\D_{(b-1,1)}(B)\leq \D^{V}_{b} (B)$ and (2) $\D^E_{a+b+1}(G)\leq \D^E_{a}(F)+\D^E_{b}(B)$ when the graphs $F$ and $B$ are $k_F$-edge connected and $k_B$-edge connected, $0\leq a < k_F$, $0\leq b < k_B$, and provided that $\D^E_{a}(F)\geq 2$ and $\D^E_{b}(B)\geq 2$.