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Kosowski, A., Li, B., Nisse, N., & Suchan, K. (2015). kChordal Graphs: From Cops and Robber to Compact Routing via Treewidth. Algorithmica, 72(3), 758–777.
Abstract: Cops and robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43(23), 235239, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38(1), 8992, 1985), concern a team of cops that must capture a robber moving in a graph. We consider the class of kchordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than k, ka parts per thousand yen3. We prove that k1 cops are always sufficient to capture a robber in kchordal graphs. This leads us to our main result, a new structural decomposition for a graph class including kchordal graphs. We present a polynomialtime algorithm that, given a graph G and ka parts per thousand yen3, either returns an induced cycle larger than k in G, or computes a treedecomposition of G, each bag of which contains a dominating path with at most k1 vertices. This allows us to prove that any kchordal graph with maximum degree Delta has treewidth at most (k1)(Delta1)+2, improving the O(Delta(Delta1) (k3)) bound of Bodlaender and Thilikos (Discrete Appl. Math. 79(13), 4561, 1997. Moreover, any graph admitting such a treedecomposition has small hyperbolicity). As an application, for any nvertex graph admitting such a treedecomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(klog Delta+logn) bits and achieving an additive stretch of O(klog Delta). As far as we know, this is the first routing scheme with O(klog Delta+logn)routing tables and small additive stretch for kchordal graphs.
Keywords: Treewidth; Chordality; Compact routing; Cops and robber games
