600 research outputs found
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
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
Cops and Invisible Robbers: the Cost of Drunkenness
We examine a version of the Cops and Robber (CR) game in which the robber is
invisible, i.e., the cops do not know his location until they capture him.
Apparently this game (CiR) has received little attention in the CR literature.
We examine two variants: in the first the robber is adversarial (he actively
tries to avoid capture); in the second he is drunk (he performs a random walk).
Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD),
which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being
the expected capture times in the adversarial and drunk CiR variants,
respectively. We show that these capture times are well defined, using game
theory for the adversarial case and partially observable Markov decision
processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD
for several special graph families such as -regular trees, give some bounds
for grids, and provide general upper and lower bounds for general classes of
graphs. We also give an infinite family of graphs showing that iCOD can be
arbitrarily close to any value in [2,infinty). Finally, we briefly examine one
more CiR variant, in which the robber is invisible and "infinitely fast"; we
argue that this variant is significantly different from the Graph Search game,
despite several similarities between the two games
A note on the Cops & Robber game on graphs embedded in non-orientable surfaces
The Cops and Robber game is played on undirected finite graphs. A number of
cops and one robber are positioned on vertices and take turns in sliding along
edges. The cops win if they can catch the robber. The minimum number of cops
needed to win on a graph is called its cop number. It is known that the cop
number of a graph embedded on a surface of genus is at most ,
if is orientable (Schroeder 2004), and at most , otherwise
(Nowakowski & Schroeder 1997).
We improve the bounds for non-orientable surfaces by reduction to the
orientable case using covering spaces.
As corollaries, using Schroeder's results, we obtain the following: the
maximum cop number of graphs embeddable in the projective plane is 3; the cop
number of graphs embeddable in the Klein Bottle is at most 4, and an upper
bound is for all other .Comment: 5 pages, 1 figur
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
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