The fault-diameter of Cartesian products
Janez Žerovnik and Iztok Banič
Advances in applied mathematics Volume 40, Number 1, pp. 98-106, 2008. ISSN 0196-8858

## Abstract

Let $G$ be a $k$-connected graph and ${\mathcal{D}}_C(G)$ denote the maximum diameter of $G$ after deleting any of its $c<k$ vertices. We prove that if $G_1, G_2,\dots ,G_q$ are $k_1$-connected, $k_2$-connected,..., $k_q$-connected graphs and $0 \le a_1 < k_1$, $0 \le a_2 < k_2$,..., $0 \le a_q < k_q$ and $a = a_1 + a_2 + \dots + a_q + (q-1)$, then the fault diameter of $G$, the Cartesian product of $G_1, G_2, \dots, G_q$, with a faulty nodes satisfies the inequality ${\mathcal{D}}_a(G) \le {\mathcal{D}}_{a_1}(G_1) + {\mathcal{D}}_{a_2}(G_2) + \dots + {\mathcal{D}}_{a_q}(G_q) + 1$.

EPrint Type: Article Project Keyword UNSPECIFIED Theory & Algorithms 4606 Igor Pesek 13 March 2009