hal-00671861, version 4

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

Adrian Kosowski (, ) 1, Bi Li () 2, Nicolas Nisse () 2, Karol Suchan () 3

N° RR-7888 (2012)

Résumé : {\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 hyperbolicity $\leq\lfloor \frac{3}{2}k\rfloor$. As an application, for any $n$-node graph admitting such a tree-decomposition, 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.

• Domaine : Informatique/Algorithme et structure de données
Informatique/Mathématique discrète
• Mots-clés : Treewidth – chordality – hyperbolicity – compact routing – cops and robber games
• Référence interne : RR-7888
• Versions disponibles :  v1 (19-02-2012) v2 (19-02-2012) v3 (20-02-2012) v4 (27-02-2012)

• hal-00671861, version 4
• oai:hal.archives-ouvertes.fr:hal-00671861
• Contributeur :
• Soumis le : Vendredi 24 Février 2012, 16:16:21
• Dernière modification le : Mercredi 29 Février 2012, 16:21:11