68 research outputs found
Cops and Robbers on diameter two graphs
In this short paper we study the game of Cops and Robbers, played on the
vertices of some fixed graph of order . The minimum number of cops
required to capture a robber is called the cop number of . We show that the
cop number of graphs of diameter 2 is at most , improving a recent
result of Lu and Peng by a constant factor. We conjecture that this bound is
still not optimal, and obtain some partial results towards the optimal bound.Comment: 5 page
Lower Bounds for the Cop Number When the Robber is Fast
We consider a variant of the Cops and Robbers game where the robber can move
t edges at a time, and show that in this variant, the cop number of a d-regular
graph with girth larger than 2t+2 is Omega(d^t). By the known upper bounds on
the order of cages, this implies that the cop number of a connected n-vertex
graph can be as large as Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This
improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh
(Variations on Cops and Robbers, J. Graph Theory, 2011) when 1<t<7. We also
conjecture a general upper bound O(n^{t/t+1}) for the cop number in this
variant, generalizing Meyniel's conjecture.Comment: 5 page
Adapting Search Theory to Networks
The CSE is interested in the general problem of locating objects in networks. Because of their exposure to search theory, the problem they brought to the workshop was phrased in terms of adapting search theory to networks. Thus, the first step was the introduction of an already existing healthy literature on searching graphs.
T. D. Parsons, who was then at Pennsylvania State University, was approached in 1977 by some local spelunkers who asked his aid in optimizing a search for someone lost in a cave in Pennsylvania. Parsons quickly formulated the problem as a search problem in a graph. Subsequent papers led to two divergent problems. One problem dealt with searching under assumptions of fairly extensive information, while the other problem dealt with searching under assumptions of essentially zero information. These two topics are developed in the next two sections
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
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].
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