11,551 research outputs found
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
Cop and robber games when the robber can hide and ride
International audienceIn the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G = (V , E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski, Winkler (1983) and Quilliot (1983) characterized the cop-win graphs as dismantlable graphs. In this talk, we will characterize in a similar way the class CWFR(s, s′ ) of cop-win graphs in the game in which the cop and the robber move at different speeds s′ and s, s′ ≤ s. We also establish some connections between cop-win graphs for this game with s′ 1. We characterize the graphs which are k-winnable for any value of k
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
Throttling for the game of Cops and Robbers on graphs
We consider the cop-throttling number of a graph for the game of Cops and
Robbers, which is defined to be the minimum of , where
is the number of cops and is the minimum number of
rounds needed for cops to capture the robber on over all possible
games. We provide some tools for bounding the cop-throttling number, including
showing that the positive semidefinite (PSD) throttling number, a variant of
zero forcing throttling, is an upper bound for the cop-throttling number. We
also characterize graphs having low cop-throttling number and investigate how
large the cop-throttling number can be for a given graph. We consider trees,
unicyclic graphs, incidence graphs of finite projective planes (a Meyniel
extremal family of graphs), a family of cop-win graphs with maximum capture
time, grids, and hypercubes. All the upper bounds on the cop-throttling number
we obtain for families of graphs are .Comment: 22 pages, 4 figure
Cops, robbers, and infinite graphs
Cops and robbers is a game between two players, where one tries to catch the
other by moving along the edges of a graph. It is well known that on a finite
graph the cop has a winning strategy if and only if the graph is constructible
and that finiteness is necessary for this result.
We propose the notion of weakly cop-win graphs, a winning criterion for
infinite graphs which could lead to a generalisation. In fact, we generalise
one half of the result, that is, we prove that every constructible graph is
weakly cop-win. We also show that a similar notion studied by Chastand et al.
(which they also dubbed weakly cop-win) is not sufficient to generalise the
above result to infinite graphs.
In the locally finite case we characterise the constructible graphs as the
graphs for which the cop has a so-called protective strategy and prove that the
existence of such a strategy implies constructibility even for non-locally
finite graphs
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