abstract: 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>2. 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>2, either returns an induced cycle larger thank 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 D has treewidth at most (k-1)(D-1)+2, improving the O(D(D-1)^{k-3}) bound of Bodlaender and Thilikos (1997). Moreover, any graph admitting such a tree-decomposition has hyperbolicity at most 3k/2. As an application, for any n-node graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(log n) bits and achieving an additive stretch of O(k log D). 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.

abstract: The Autonomous System (AS) topology of the Internet (up to 61k ASs) is growing at a rate of about 10\% per year. The Border Gateway Protocol (BGP) starts to show its limits in terms of the number of routing table entries it can dynamically process and control. Due to the increasing routing information processing and storage, the same trend is observed for routing model simulators such as DRMSim specialized in large-scale simulations of routing models. Therefore, DRMSim needs enhancements to support the current size of the Internet topology and its evolution (up to 100k ASs). To this end, this paper proposes a feasibility study of the extension of DRMSim so as to support the Distributed Parallel Discrete Event paradigm. We first detail the possible distribution models and their associated communication overhead. Then, we analyze this overhead by executing BGP on a partitioned topology according to different scenarios. Finally, we conclude on the feasibility of such a simulator by computing the expected additional time required by a distributed simulation of BGP compared to its sequential simulation.