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

Adrian Kosowski 1, 2 Bi Li 3, 4 Nicolas Nisse 4 Karol Suchan 5, 6
2 GANG - Networks, Graphs and Algorithms
LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications, Inria Paris-Rocquencourt
4 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Cops and robber games, introduced by Winkler and Nowakowski [41] and independently defined by Quilliot [43], 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 (chordless) cycle of length greater than k, k ≥ 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 polynomial-time algorithm that, given a graph G and k ≥ 3, either returns an induced cycle larger than k in G, or computes a tree-decomposition of G, each 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 ∆ has treewidth at most (k −1)(∆ −1) +2, improving the O(∆ (∆ −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-vertex graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(k log ∆ + log n) bits and achieving an additive stretch of O(k log ∆). As far as we know, this is the first routing scheme with O(k log ∆ + log n)-routing tables and small additive stretch for k-chordal graphs.
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Submitted on : Saturday, June 13, 2015 - 10:51:25 AM
Last modification on : Tuesday, December 8, 2020 - 10:24:04 AM
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Adrian Kosowski, Bi Li, Nicolas Nisse, Karol Suchan. k-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth. Algorithmica, Springer Verlag, 2015, 72 (3), pp.758-777. ⟨hal-01163494⟩



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