103 research outputs found
Localization game on geometric and planar graphs
The main topic of this paper is motivated by a localization problem in
cellular networks. Given a graph we want to localize a walking agent by
checking his distance to as few vertices as possible. The model we introduce is
based on a pursuit graph game that resembles the famous Cops and Robbers game.
It can be considered as a game theoretic variant of the \emph{metric dimension}
of a graph. We provide upper bounds on the related graph invariant ,
defined as the least number of cops needed to localize the robber on a graph
, for several classes of graphs (trees, bipartite graphs, etc). Our main
result is that, surprisingly, there exists planar graphs of treewidth and
unbounded . On a positive side, we prove that is bounded
by the pathwidth of . We then show that the algorithmic problem of
determining is NP-hard in graphs with diameter at most .
Finally, we show that at most one cop can approximate (arbitrary close) the
location of the robber in the Euclidean plane
Meyniel's conjecture holds for random graphs
In the game of cops and robber, the cops try to capture a robber moving on
the vertices of the graph. The minimum number of cops required to win on a
given graph is called the cop number of . The biggest open conjecture in
this area is the one of Meyniel, which asserts that for some absolute constant
, the cop number of every connected graph is at most .
In this paper, we show that Meyniel's conjecture holds asymptotically almost
surely for the binomial random graph. We do this by first showing that the
conjecture holds for a general class of graphs with some specific
expansion-type properties. This will also be used in a separate paper on random
-regular graphs, where we show that the conjecture holds asymptotically
almost surely when .Comment: revised versio
Almost all cop-win graphs contain a universal vertex
AbstractWe consider cop-win graphs in the binomial random graph G(n,1/2). We prove that almost all cop-win graphs contain a universal vertex. From this result, we derive that the asymptotic number of labelled cop-win graphs of order n is equal to (1+o(1))n2n2/2−3n/2+1
Revolutionaries and spies on random graphs
Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a
simplified model for network security. In the game we consider in this paper, a
team of revolutionaries tries to hold an unguarded meeting consisting of
revolutionaries. A team of spies wants to prevent this forever. For
given and , the minimum number of spies required to win on a graph
is the spy number . We present asymptotic results for the game
played on random graphs for a large range of , and
. The behaviour of the spy number is analyzed completely for dense
graphs (that is, graphs with average degree at least n^{1/2+\eps} for some
\eps > 0). For sparser graphs, some bounds are provided
Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity
We call a (simple) graph G codismantlable if either it has no edges or else
it has a codominated vertex x, meaning that the closed neighborhood of x
contains that of one of its neighbor, such that G-x codismantlable. We prove
that if G is well-covered and it lacks induced cycles of length four, five and
seven, than the vertex decomposability, codismantlability and
Cohen-Macaulayness for G are all equivalent. The rest deals with the
computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note
that our approach complements and unifies many of the earlier results on
bipartite, chordal and very well-covered graphs
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