225 research outputs found

    Locating a robber with multiple probes

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    We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any nn-vertex graph GG there is a winning strategy for the cop on the graph G1/mG^{1/m} obtained by replacing each edge of GG by a path of length mm, if mnm\geq n. The present authors showed that, for all but a few small values of nn, this bound may be improved to mn/2m\geq n/2, which is best possible. In this paper we consider the natural extension in which the cop probes a set of kk vertices, rather than a single vertex, at each turn. We consider the relationship between the value of kk required to ensure victory on the original graph and the length of subdivisions required to ensure victory with k=1k=1. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of kk for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree Δ\Delta, which is best possible up to a factor of (1o(1))(1-o(1)).Comment: 16 pages, 2 figures. Updated to show that Theorem 2 also applies to infinite graphs. Accepted for publication in Discrete Mathematic

    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

    Centroidal localization game

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    One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph GG is the minimum number kk of detectors placed in some vertices {v1,,vk}\{v_1,\cdots,v_k\} such that the vector (d1,,dk)(d_1,\cdots,d_k) of the distances d(vi,r)d(v_i,r) between the detectors and the entity's location rr allows to uniquely determine rV(G)r \in V(G). In a more realistic setting, instead of getting the exact distance information, given devices placed in {v1,,vk}\{v_1,\cdots,v_k\}, we get only relative distances between the entity's location rr and the devices (for every 1i,jk1\leq i,j\leq k, it is provided whether d(vi,r)>d(v_i,r) >, <<, or == to d(vj,r)d(v_j,r)). The centroidal dimension of a graph GG is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set {v1,,vk}\{v_1,\cdots,v_k\} of vertices is probed and then the relative distances between the vertices viv_i and the current location rr of the entity are given. If not located, the moving entity may move along one edge. Let ζ(G)\zeta^* (G) be the minimum kk such that the entity is eventually located, whatever it does, in the graph GG. We prove that ζ(T)2\zeta^* (T)\leq 2 for every tree TT and give an upper bound on ζ(GH)\zeta^*(G\square H) in cartesian product of graphs GG and HH. Our main result is that ζ(G)3\zeta^* (G)\leq 3 for any outerplanar graph GG. We then prove that ζ(G)\zeta^* (G) is bounded by the pathwidth of GG plus 1 and that the optimization problem of determining ζ(G)\zeta^* (G) is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn

    Subdivisions in the Robber Locating Game

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    We consider a game in which a cop searches for a moving robber on a graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph GG there is a winning strategy for the cop on the graph G1/mG^{1/m} obtained by replacing each edge of GG by a path of length mm, if mnm \geqslant n. They conjectured that this bound was best possible for complete graphs, but the present authors showed that in fact the cop wins on K1/mK^{1/m} if and only if mn/2m \geqslant n/2, for all but a few small values of nn. In this paper we extend this result to general graphs by proving that the cop has a winning strategy on G1/mG^{1/m} provided mn/2m \geqslant n/2 for all but a few small values of nn; this bound is best possible. We also consider replacing the edges of GG with paths of varying lengths.Comment: 13 Page

    Subgraphs and Colourability of Locatable Graphs

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    We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a different direction of topic, we also start research in the direction of colourability of locatable graphs, we also show that every locatable graph is 4-colourable, but not necessarily 3-colourable.Comment: 25 page

    Searching and Sorting Algorithms

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    This dissertation analyses two combinatorial questions that involve algorithmic solutions. First we consider the Robber Locating Game, a pursuit-evasion game introduced by Seager in 2012. This game is a variant of the renowned Cops and Robbers game; in this variant the robber does not disclose his location to the cop, and her aim is merely to locate rather than capture him. Although he moves around the graph as normal on his turns, on her turns she picks any vertex freely and asks how far he is from her probed vertex. We call a graph locatable if there is a possible cop strategy that will always locate the robber in finitely many moves, and non-locatable otherwise.In this dissertation we consider how much subdivision of a graph is necessary to make it locatable, establishing exact bounds in the case of complete and complete bipartite graphs, and a general (n/2 + 1) bound for all finite graphs. We also consider subdividing infinite graphs, exhibiting a sufficient subdivision function for the cases where subdividing them can make them locatable. Finally we close with a series of results about the game, including the relationship between locatability number and maximum degree and showing that every locatable graph is 4-colourable.In the second part we consider how a user can determine the ordering of a well-ordered set of elements, when he initially does not know the ordering but is given a scale. This scale takes k elements and returns the t_1, t_2, ..., t_s of them according to this ordering. We show that he cannot determine the complete ordering, since he cannot order the initial and final segments. Apart from this restriction we outline algorithms to enable the user to determine the ordering in both the online and offline cases. We show that in the online case he can determine the ordering in O(n log n) queries, and in the offline case in O(n^{k-t+1}) queries, which we show is the best possible order of the number of queries
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