# k-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth

1 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms
Université Sciences et Technologies - Bordeaux 1, Inria Bordeaux - Sud-Ouest, École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), CNRS - Centre National de la Recherche Scientifique : UMR5800
3 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : {\it Cops and robber games} concern a team of cops that must capture a robber moving in a graph. We consider the class of $k$-chordal graphs, i.e., graphs with no induced cycle of length greater than $k$, $k\geq 3$. We prove that $k-1$ cops are always sufficient to capture a robber in $k$-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including $k$-chordal graphs. We present a quadratic algorithm that, given a graph $G$ and $k\geq 3$, either returns an induced cycle larger than $k$ in $G$, or computes a {\it tree-decomposition} of $G$, each {\it bag} of which contains a dominating path with at most $k-1$ vertices. This allows us to prove that any $k$-chordal graph with maximum degree $\Delta$ has treewidth at most $(k-1)(\Delta-1)+2$, improving the $O(\Delta (\Delta-1)^{k-3})$ bound of Bodlaender and Thilikos (1997). Moreover, any graph admitting such a tree-decomposition has small hyperbolicity. As an application, for any $n$-node graph admitting such a tree-decomposi\-tion, we propose a {\it compact routing scheme} using routing tables, addresses and headers of size $O(\log n)$ bits and achieving an additive stretch of $O(k\log \Delta)$. As far as we know, this is the first routing scheme with $O(\log n)$-routing tables and small additive stretch for $k$-chordal graphs.
Type de document :
Communication dans un congrès
ICALP - 39th International Colloquium on Automata, Languages and Programming, track C, 2012, United Kingdom. pp.610-622, 2012
Domaine :

https://hal.archives-ouvertes.fr/hal-00704202
Contributeur : Nicolas Nisse <>
Soumis le : lundi 4 juin 2012 - 23:06:17
Dernière modification le : vendredi 11 septembre 2015 - 01:07:08
Document(s) archivé(s) le : jeudi 15 décembre 2016 - 10:44:31

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kchordalICALPfinal.pdf
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• HAL Id : hal-00704202, version 1

### Citation

Adrian Kosowski, Bi Li, Nicolas Nisse, Karol Suchan. k-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth. ICALP - 39th International Colloquium on Automata, Languages and Programming, track C, 2012, United Kingdom. pp.610-622, 2012. <hal-00704202>

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