# Minimum Sizes of Identifying Codes in Graphs Differing by One Edge or One Vertex

Abstract : Let $G$ be a simple, undirected graph with vertex set $V$. For $v \in V$ and $r \geq 1$, we denote by $B_{G,r}(v)$ the ball of radius $r$ and centre $v$. A set $C \subseteq V$ is said to be an $r$-identifying code in $G$ if the sets $B_{G,r}(v) \cap C$, $v \in V$ , are all nonempty and distinct. A graph $G$ admitting an $r$- identifying code is called $r$-twin-free, and in this case the size of a smallest $r$-identifying code in $G$ is denoted by $\gamma_r(G)$. We study the following structural problem: let $G$ be an $r$-twin-free graph, and $G^*$ be a graph obtained from $G$ by adding or deleting a vertex, or by adding or deleting an edge. If $G^*$ is still $r$-twin-free, we compare the behaviours of $\gamma_r(G)$ and $\gamma_r(G^*)$, establishing results on their possible differences and ratios.
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https://hal.archives-ouvertes.fr/hal-00731811
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Submitted on : Thursday, September 13, 2012 - 4:09:15 PM
Last modification on : Monday, January 24, 2022 - 11:43:20 AM
Long-term archiving on: : Friday, December 14, 2012 - 3:57:44 AM

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• HAL Id : hal-00731811, version 1

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Irène Charon, Iiro Honkala, Olivier Hudry, Antoine Lobstein. Minimum Sizes of Identifying Codes in Graphs Differing by One Edge or One Vertex. 2012. ⟨hal-00731811⟩

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