4,512 research outputs found
Cops vs. Gambler
We consider a variation of cop vs.\ robber on graph in which the robber is
not restricted by the graph edges; instead, he picks a time-independent
probability distribution on and moves according to this fixed
distribution. The cop moves from vertex to adjacent vertex with the goal of
minimizing expected capture time. Players move simultaneously. We show that
when the gambler's distribution is known, the expected capture time (with best
play) on any connected -vertex graph is exactly . We also give bounds on
the (generally greater) expected capture time when the gambler's distribution
is unknown to the cop.Comment: 6 pages, 0 figure
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].
On Necessary and Sufficient Number of Cops in the Game of Cops and Robber in Multidimensional Grids
We theoretically analyze the Cops and Robber Game for the first time in a
multidimensional grid. It is shown that for an -dimensional grid, at least
cops are necessary to ensure capture of the robber. We also present a set
of cop strategies for which cops are provably sufficient to catch the
robber. Further, for two-dimensional grid, we provide an efficient cop strategy
for which the robber is caught even by a single cop under certain conditions.Comment: This is a revised and extended version of the poster paper with the
same title that has been presented in the 8th Asian Symposium on Computer
Mathematics (ASCM), December 15-17, 2007, Singapor
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