390 research outputs found
Subdivisions in the Robber Locating Game
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
there is a winning strategy for the cop on the graph obtained by
replacing each edge of by a path of length , if . They
conjectured that this bound was best possible for complete graphs, but the
present authors showed that in fact the cop wins on if and only if , for all but a few small values of . In this paper we extend
this result to general graphs by proving that the cop has a winning strategy on
provided for all but a few small values of ;
this bound is best possible. We also consider replacing the edges of with
paths of varying lengths.Comment: 13 Page
Locating a robber with multiple probes
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
-vertex graph there is a winning strategy for the cop on the graph
obtained by replacing each edge of by a path of length , if
. The present authors showed that, for all but a few small values of
, this bound may be improved to , which is best possible. In this
paper we consider the natural extension in which the cop probes a set of
vertices, rather than a single vertex, at each turn. We consider the
relationship between the value of required to ensure victory on the
original graph and the length of subdivisions required to ensure victory with
. 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 for which the cop has a winning strategy on
any (possibly infinite) connected graph of maximum degree , which is
best possible up to a factor of .Comment: 16 pages, 2 figures. Updated to show that Theorem 2 also applies to
infinite graphs. Accepted for publication in Discrete Mathematic
The -visibility Localization Game
We study a variant of the Localization game in which the cops have limited
visibility, along with the corresponding optimization parameter, the
-visibility localization number , where is a non-negative
integer. We give bounds on -visibility localization numbers related to
domination, maximum degree, and isoperimetric inequalities. For all , we
give a family of trees with unbounded values. Extending results known
for the localization number, we show that for , every tree contains a
subdivision with . For many , we give the exact value of
for the Cartesian grid graphs, with the remaining cases
being one of two values as long as is sufficiently large. These examples
also illustrate that for all distinct choices of and
$j.
Searching and Sorting Algorithms
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
Sequential Metric Dimension
International audienceSeager introduced the following game in 2013. An invisible and immobile target is hidden at some vertex of a graph . Every step, one vertex of can be probed which results in the knowledge of the distance between and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is. We address the generalization of this game where vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph and two integers , the Localization Problem asks whether there exists a strategy to locate a target hidden in in at most steps by probing at most vertices per step. We show this problem is NP-complete when (resp.,) is a fixed parameter. Our main results are for the class of trees where we prove this problem is NP-complete when and are part of the input but, despite this, we design a polynomial-time (+1)-approximation algorithm in trees which gives a solution using at most one more step than the optimal one. It follows that the Localization Problem is polynomial-time solvable in trees if is fixed
Sequential Metric Dimension
International audienceIn the localization game, introduced by Seager in 2013, an invisible and immobile target is hidden at some vertex of a graph . At every step, one vertex of can be probed which results in the knowledge of the distance between and the secret location of the target. The objective of the game is to minimize the number of steps needed to locate the target whatever be its location.We address the generalization of this game where vertices can be probed at every step. Our game also generalizes the notion of the {\it metric dimension} of a graph.Precisely, given a graph and two integers , the {\sc Localization} problem asks whether there exists a strategy to locate a target hidden in in at most steps and probing at most vertices per step. We first show that, in general, this problem is \textsf{NP}-complete for every fixed (resp., ).We then focus on the class of trees.On the negative side,we prove that the \Localization problem is \textsf{NP}-complete in trees when and are part of the input. On the positive side, we design a -approximation algorithm for the problem in -node trees, {\it i.e.}, an algorithm that computes in time (independent of ) a strategy to locate the target in at most one more step than an optimal strategy. This algorithm can be used to solve the \Localization problem in trees in polynomial time if is fixed.We also consider some of these questions in the context where, upon probing the vertices,the relative distances to the target are retrieved.This variant of the problem generalizes the notion of the {\it centroidal dimension} of a graph
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