, Volume 72, Issue 3, pp 758–777 | Cite as

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

  • A. Kosowski
  • B. Li
  • N. Nisse
  • K. Suchan


Cops and robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43(2–3), 235–239, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38(1), 89–92, 1985), 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 Δ has treewidth at most (k−1)(Δ−1)+2, improving the O(Δ(Δ−1) k−3) bound of Bodlaender and Thilikos (Discrete Appl. Math. 79(1–3), 45–61, 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(klogΔ+logn) bits and achieving an additive stretch of O(klogΔ). As far as we know, this is the first routing scheme with O(klogΔ+logn)-routing tables and small additive stretch for k-chordal graphs.


Treewidth Chordality Compact routing Cops and robber games 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CEPAGEINRIA, LaBRITalenceFrance
  2. 2.Coati ProjectINRIA, I3S(CNRS/UNS)Sophia AntipolisFrance
  3. 3.Institute of Applied Mathematics, AAMSCASBeijingChina
  4. 4.FICUniversidad Adolfo IbáñezSantiagoChile
  5. 5.WMS, AGHUniversity of Science and TechnologyKrakowPoland

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