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| HAL : hal-00530172, version 3 |
| arXiv : 1010.5975 |
| DOI : 10.1016/j.dam.2012.02.009 |
| Fiche détaillée |  Récupérer au format |
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| Discrete Applied Mathematics 160, 10-11 (2012) 1532-1546 |
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| Versions disponibles : | v1 (28-10-2010) | v2 (13-06-2011) | v3 (29-06-2012) |
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| On the size of identifying codes in triangle-free graphs |
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Florent Foucaud 1Ralf Klasing 1, 2 |
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| (01/07/2012) |
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| In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let $\M(G)$ be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta\geq 3$, $\M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}$. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}{\Delta}+c$ holds for any nontrivial connected identifiable graph $G$. |
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| 1 : | Laboratoire Bordelais de Recherche en Informatique (LaBRI) |
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CNRS : UMR5800 – Université Sciences et Technologies - Bordeaux I – École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB) – Université Victor Segalen - Bordeaux II |
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| 2 : | CEPAGE (INRIA Bordeaux - Sud-Ouest) |
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INRIA – CNRS : UMR5800 – Université Sciences et Technologies - Bordeaux I – École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB) |
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| 3 : | Department of Algorithms and System Modeling [Gdansk] |
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Gdansk University of Technology |
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| Domaine | : | Informatique/Mathématique discrète Mathématiques/Combinatoire |
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| Identifying code – Dominating set – Triangle-free graph – Maximum degree |
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| hal-00530172, version 3 | |
| http://hal.archives-ouvertes.fr/hal-00530172 | |
| oai:hal.archives-ouvertes.fr:hal-00530172 | |
| Contributeur : Florent Foucaud | |
| Soumis le : Vendredi 29 Juin 2012, 17:34:52 | |
| Dernière modification le : Vendredi 29 Juin 2012, 20:57:35 | |
