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

Abstract

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.