231 research outputs found
Cops and Robbers on diameter two graphs
In this short paper we study the game of Cops and Robbers, played on the
vertices of some fixed graph of order . The minimum number of cops
required to capture a robber is called the cop number of . We show that the
cop number of graphs of diameter 2 is at most , improving a recent
result of Lu and Peng by a constant factor. We conjecture that this bound is
still not optimal, and obtain some partial results towards the optimal bound.Comment: 5 page
Hyperopic Cops and Robbers
We introduce a new variant of the game of Cops and Robbers played on graphs,
where the robber is invisible unless outside the neighbor set of a cop. The
hyperopic cop number is the corresponding analogue of the cop number, and we
investigate bounds and other properties of this parameter. We characterize the
cop-win graphs for this variant, along with graphs with the largest possible
hyperopic cop number. We analyze the cases of graphs with diameter 2 or at
least 3, focusing on when the hyperopic cop number is at most one greater than
the cop number. We show that for planar graphs, as with the usual cop number,
the hyperopic cop number is at most 3. The hyperopic cop number is considered
for countable graphs, and it is shown that for connected chains of graphs, the
hyperopic cop density can be any real number in $[0,1/2].
Visibility Graphs, Dismantlability, and the Cops and Robbers Game
We study versions of cop and robber pursuit-evasion games on the visibility
graphs of polygons, and inside polygons with straight and curved sides. Each
player has full information about the other player's location, players take
turns, and the robber is captured when the cop arrives at the same point as the
robber. In visibility graphs we show the cop can always win because visibility
graphs are dismantlable, which is interesting as one of the few results
relating visibility graphs to other known graph classes. We extend this to show
that the cop wins games in which players move along straight line segments
inside any polygon and, more generally, inside any simply connected planar
region with a reasonable boundary. Essentially, our problem is a type of
pursuit-evasion using the link metric rather than the Euclidean metric, and our
result provides an interesting class of infinite cop-win graphs.Comment: 23 page
Variations on Cops and Robbers
We consider several variants of the classical Cops and Robbers game. We treat
the version where the robber can move R > 1 edges at a time, establishing a
general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha
= 1 + 1/R, thus generalizing the best known upper bound for the classical case
R = 1 due to Lu and Peng. We also show that in this case, the cop number of an
N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N
if R is infinite. For R = 1, we study the directed graph version of the
problem, and show that the cop number of any strongly connected digraph on N
vertices is at most O(N(log log N)^2/log N). Our approach is based on
expansion.Comment: 18 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
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
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