225 research outputs found
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
Localization game on geometric and planar graphs
The main topic of this paper is motivated by a localization problem in
cellular networks. Given a graph 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 ,
defined as the least number of cops needed to localize the robber on a graph
, for several classes of graphs (trees, bipartite graphs, etc). Our main
result is that, surprisingly, there exists planar graphs of treewidth and
unbounded . On a positive side, we prove that is bounded
by the pathwidth of . We then show that the algorithmic problem of
determining is NP-hard in graphs with diameter at most .
Finally, we show that at most one cop can approximate (arbitrary close) the
location of the robber in the Euclidean plane
Centroidal localization game
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 is the minimum number of detectors placed
in some vertices such that the vector
of the distances between the detectors and the entity's location
allows to uniquely determine . In a more realistic setting, instead
of getting the exact distance information, given devices placed in
, we get only relative distances between the entity's
location and the devices (for every , it is provided
whether , , or to ). The centroidal dimension of a
graph 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 of vertices is probed and then the relative distances
between the vertices and the current location of the entity are
given. If not located, the moving entity may move along one edge. Let be the minimum such that the entity is eventually located, whatever it
does, in the graph .
We prove that for every tree and give an upper bound
on in cartesian product of graphs and . Our main
result is that for any outerplanar graph . We then prove
that is bounded by the pathwidth of plus 1 and that the
optimization problem of determining 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
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
Subgraphs and Colourability of Locatable Graphs
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
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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