Edge, vertex and mixed fault diameters.
Janez Žerovnik, Iztok Banič and Rija Erveš
Adv. appl. math. Volume 43, Number 3, pp. 231-238, 2009.

Abstract

Let ${\mathcal{D}}^E_q(G)$ denote the maximum diameter among all subgraphs obtained by deleting $q$ edges of $G$. Let ${\mathcal{D}}^V_p(G)$ denote the maximum diameter among all subgraphs obtained by deleting $p$ vertices of $G$. We prove that ${\mathcal{D}}^E_a(G) \leqslant {\mathcal{D}}^V_a(G) + 1$ a for all meaningful $a$. We also define mixed fault diameter ${\mathcal{D}}^M_{(p,q)}(G)$, where $p$ vertices and $q$ edges are deleted at the same time. We prove that for $0 < l \leqslant a$, ${\mathcal{D}}^E_a(G) \leqslant {\mathcal{D}}^M_{(a-\ell,\ell)}(G) \leqslant {\mathcal{D}}^V_a(G) + 1$, and give some examples.

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