Mixed fault diameter of Cartesian graph bundles II

Abstract

The mixed fault diameter D (p,q) (G) is the maximum diameter among all subgraphs obtained from graph G by deleting p vertices and q edges. A graph is (p, q)+connected if it remains connected after removal of any p vertices and any q edges. Let F be a connected graph with the diameter D(F ) > 1, and B be (p, q)+connected graph. Upper bounds for the mixed fault diameter of Cartesian graph bundle G with fibre F over the base graph B are given. We prove that if q > 0, then

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

“…Upper bounds for the mixed fault diameter of Cartesian graph bundles are given in [15,16] that in some case also improve previously known results on vertex and edge fault diameters on these classes of Cartesian graph bundles [2,5]. However results in [15] address only the number of faults given by the connectivity of the fibre (plus one vertex), while the connectivity of the graph bundle can be much higher when the connectivity of the base graph is substantial, and results in [16] address only the number of faults given by the connectivity of the base graph (plus one vertex), while the connectivity of the graph bundle can be much higher when the connectivity of the fibre is substantial. An upper bound for the mixed fault diameter that would take into account both types of faults remains to be an interesting open research problem.…”

“…In recent work [15,16], an upper bound for the mixed fault diameter of Cartesian graph bundles, D (p+1,q) (G), in terms of mixed fault diameter of the fibre and diameter of the base graph and in terms of diameter of the fibre and mixed fault diameter of the base graph, respectively, is given.…”

“…Theorem 4.5 [15] Let G be a Cartesian graph bundle with fibre F over the base graph B, where graph F is (p, q)+connected, p + q > 0, and B is a connected graph with diameter D(B) > 1. Then we have: Theorem 4.6 [16] Let G be a Cartesian graph bundle with fibre F over the base graph B, graph F be a connected graph with diameter D(F ) > 1, and graph B be (p, q)+connected, p + q > 0. Then we have:…”

“…Methods used involve the theory of mixed connectivity and recent results on mixed fault diameters [6,14,15,16]. For completeness, we also give the analogous improved upper bound for edge fault diameter.…”