418 research outputs found

    Almost all cop-win graphs contain a universal vertex

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    AbstractWe consider cop-win graphs in the binomial random graph G(n,1/2). We prove that almost all cop-win graphs contain a universal vertex. From this result, we derive that the asymptotic number of labelled cop-win graphs of order n is equal to (1+o(1))n2n2/2−3n/2+1

    Localization game on geometric and planar graphs

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    The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph GG we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant ζ(G)\zeta (G), defined as the least number of cops needed to localize the robber on a graph GG, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth 22 and unbounded ζ(G)\zeta (G). On a positive side, we prove that ζ(G)\zeta (G) is bounded by the pathwidth of GG. We then show that the algorithmic problem of determining ζ(G)\zeta (G) is NP-hard in graphs with diameter at most 22. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

    Meyniel's conjecture holds for random graphs

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    In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph GG is called the cop number of GG. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant CC, the cop number of every connected graph GG is at most C∣V(G)∣C \sqrt{|V(G)|}. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random dd-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=d(n)≄3d = d(n) \ge 3.Comment: revised versio

    On the minimum order of kk-cop win graphs

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    We consider the minimum order graphs with a given cop number. We prove that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. We provide the results of a computational search on the cop number of all graphs up to and including order 10. A relationship is presented between the minimum order of graph with cop number kk and Meyniel's conjecture on the asymptotic maximum value of the cop number of a connected graph

    Dagstuhl Reports : Volume 1, Issue 2, February 2011

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    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Bucolic Complexes

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    We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.Comment: 45 pages, 4 figure
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