k-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Reports Year : 2012

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

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 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.
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Dates and versions

hal-00671861 , version 1 (19-02-2012)
hal-00671861 , version 2 (19-02-2012)
hal-00671861 , version 3 (19-02-2012)
hal-00671861 , version 4 (24-02-2012)

Identifiers

  • HAL Id : hal-00671861 , version 4

Cite

Adrian Kosowski, Bi Li, Nicolas Nisse, Karol Suchan. k-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth. INRIA. 2012. ⟨hal-00671861v4⟩
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