104,238 research outputs found
The capture time of grids
We consider the game of Cops and Robber played on the Cartesian product of
two trees. Assuming the players play perfectly, it is shown that if there are
two cops in the game, then the length of the game (known as the 2-capture time
of the graph) is equal to half the diameter of the graph. In particular, the
2-capture time of the m x n grid is proved to be floor ((m+n-2)/2).Comment: 7 page
On a Bounded Budget Network Creation Game
We consider a network creation game in which each player (vertex) has a fixed
budget to establish links to other players. In our model, each link has unit
price and each agent tries to minimize its cost, which is either its local
diameter or its total distance to other players in the (undirected) underlying
graph of the created network. Two versions of the game are studied: in the MAX
version, the cost incurred to a vertex is the maximum distance between the
vertex and other vertices, and in the SUM version, the cost incurred to a
vertex is the sum of distances between the vertex and other vertices. We prove
that in both versions pure Nash equilibria exist, but the problem of finding
the best response of a vertex is NP-hard. We take the social cost of the
created network to be its diameter, and next we study the maximum possible
diameter of an equilibrium graph with n vertices in various cases. When the sum
of players' budgets is n-1, the equilibrium graphs are always trees, and we
prove that their maximum diameter is Theta(n) and Theta(log n) in MAX and SUM
versions, respectively. When each vertex has unit budget (i.e. can establish
link to just one vertex), the diameter of any equilibrium graph in either
version is Theta(1). We give examples of equilibrium graphs in the MAX version,
such that all vertices have positive budgets and yet the diameter is
Omega(sqrt(log n)). This interesting (and perhaps counter-intuitive) result
shows that increasing the budgets may increase the diameter of equilibrium
graphs and hence deteriorate the network structure. Then we prove that every
equilibrium graph in the SUM version has diameter 2^O(sqrt(log n)). Finally, we
show that if the budget of each player is at least k, then every equilibrium
graph in the SUM version is k-connected or has diameter smaller than 4.Comment: 28 pages, 3 figures, preliminary version appeared in SPAA'1
The domination game played on diameter 2 graphs
Let be the game domination number of a graph . It is proved
that if , then . The bound is
attained: if and , then if and only if is one of seven
sporadic graphs with or the Petersen graph, and there are exactly
ten graphs of diameter and order that attain the bound
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
Subgraphs and Colourability of Locatable Graphs
We study a game of pursuit and evasion introduced by Seager in 2012, in which
a cop searches the robber from outside the graph, using distance queries. A
graph on which the cop wins is called locatable. In her original paper, Seager
asked whether there exists a characterisation of the graph property of
locatable graphs by either forbidden or forbidden induced subgraphs, both of
which we answer in the negative. We then proceed to show that such a
characterisation does exist for graphs of diameter at most 2, stating it
explicitly, and note that this is not true for higher diameter. Exploring a
different direction of topic, we also start research in the direction of
colourability of locatable graphs, we also show that every locatable graph is
4-colourable, but not necessarily 3-colourable.Comment: 25 page
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
- âŠ