1,888 research outputs found
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
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 , 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
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
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
Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary
In a pursuit-evasion game, a team of pursuers attempt to capture an evader.
The players alternate turns, move with equal speed, and have full information
about the state of the game. We consider the most restictive capture condition:
a pursuer must become colocated with the evader to win the game. We prove two
general results about pursuit-evasion games in topological spaces. First, we
show that one pursuer has a winning strategy in any CAT(0) space under this
restrictive capture criterion. This complements a result of Alexander, Bishop
and Ghrist, who provide a winning strategy for a game with positive capture
radius. Second, we consider the game played in a compact domain in Euclidean
two-space with piecewise analytic boundary and arbitrary Euler characteristic.
We show that three pursuers always have a winning strategy by extending recent
work of Bhadauria, Klein, Isler and Suri from polygonal environments to our
more general setting.Comment: 21 pages, 6 figure
- …