475 research outputs found
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
A probabilistic version of the game of Zombies and Survivors on graphs
We consider a new probabilistic graph searching game played on graphs,
inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a
set of zombies attempts to eat a lone survivor loose on a given graph. The
zombies randomly choose their initial location, and during the course of the
game, move directly toward the survivor. At each round, they move to the
neighbouring vertex that minimizes the distance to the survivor; if there is
more than one such vertex, then they choose one uniformly at random. The
survivor attempts to escape from the zombies by moving to a neighbouring vertex
or staying on his current vertex. The zombies win if eventually one of them
eats the survivor by landing on their vertex; otherwise, the survivor wins. The
zombie number of a graph is the minimum number of zombies needed to play such
that the probability that they win is strictly greater than 1/2. We present
asymptotic results for the zombie numbers of several graph families, such as
cycles, hypercubes, incidence graphs of projective planes, and Cartesian and
toroidal grids
Graphs with Large Girth and Small Cop Number
In this paper we consider the cop number of graphs with no, or few, short
cycles. We show that when the girth of is at least and the minimum
degree is sufficiently large, where
, then as where . This extends
work of Frankl and implies that if is large and dense in the sense that
while also having girth , then
satisfies Meyniel's conjecture, that is . Moreover, it
implies that if is large and dense in the sense that there for some , while also having girth , then
there exists an such that , thereby
satisfying the weak Meyniel's conjecture. Of course, this implies similar
results for dense graphs with small, that is , numbers of
short cycles, as each cycle can be broken by adding a single cop. We also, show
that there are graphs with girth and minimum degree such that
the cop number is at most . This
resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by
showing that the constant cannot be improved in the exponent of a
lower bound .Comment: 7 pages, 0 figures, 0 table
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