63 research outputs found
Graph Searching Games and Width Measures for Directed Graphs
In cops and robber games a number of cops tries to capture a robber in
a graph. A variant of these games on undirected graphs characterises tree width by the least number of cops needed to win. We consider cops and robber games on digraphs and width measures (such as DAG-width, directed tree width or D-width) corresponding to them. All of them generalise tree width and the game characterising it.
For the DAG-width game we prove that the problem to decide the minimal
number of cops required to capture the robber (which is the same as deciding DAG-width), is PSPACE-complete, in contrast to most other similar games. We also show that the cop-monotonicity cost for directed tree width games cannot be bounded by any function. As a consequence, D-width is not bounded in directed tree width, refuting a conjecture by Safari.
A large number of directed width measures generalising tree width has been proposed in the literature. However, only very little was known about the relation between them, in particular about whether classes of digraphs of bounded width in one measure have bounded width in another. In this paper we establish an almost complete order among the most prominent width measures with respect to mutual boundedness
Graph Searching, Parity Games and Imperfect Information
We investigate the interrelation between graph searching games and games with
imperfect information. As key consequence we obtain that parity games with
bounded imperfect information can be solved in PTIME on graphs of bounded
DAG-width which generalizes several results for parity games on graphs of
bounded complexity. We use a new concept of graph searching where several cops
try to catch multiple robbers instead of just a single robber. The main
technical result is that the number of cops needed to catch r robbers
monotonously is at most r times the DAG-width of the graph. We also explore
aspects of this new concept as a refinement of directed path-width which
accentuates its connection to the concept of imperfect information
Recontamination Helps a Lot to Hunt a Rabbit
The Hunters and Rabbit game is played on a graph G where the Hunter player shoots at k vertices in every round while the Rabbit player occupies an unknown vertex and, if it is not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number h(G) of a graph G is the minimum integer k such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing h(G) remains open in general graphs and even in more restricted graph classes such as trees. To progress further in this study, we propose a notion of monotonicity (a well-studied and useful property in classical pursuit-evasion games such as Graph Searching games) for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot "must not host the rabbit anymore". This allows us to obtain new results in various graph classes.
More precisely, let the monotone hunter number mh(G) of a graph G be the minimum integer k such that the Hunter player has a monotone winning strategy. We show that pw(G) ? mh(G) ? pw(G)+1 for any graph G with pathwidth pw(G), which implies that computing mh(G), or even approximating mh(G) up to an additive constant, is NP-hard. Then, we show that mh(G) can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between h and mh, i.e., that monotonicity does not help. In particular, we show that, for every k ? 3, there exists a tree T with h(T) = 2 and mh(T) = k. We conclude by proving that computing h (resp., mh) is FPT parameterised by the minimum size of a vertex cover
Further results on the Hunters and Rabbit game through monotonicity
Hunters and Rabbit game is played on a graph where the Hunter player
shoots at vertices in every round while the Rabbit player occupies an
unknown vertex and, if not shot, must move to a neighbouring vertex after each
round. The Rabbit player wins if it can ensure that its position is never shot.
The Hunter player wins otherwise. The hunter number of a graph is
the minimum integer such that the Hunter player has a winning strategy
(i.e., allowing him to win whatever be the strategy of the Rabbit player). This
game has been studied in several graph classes, in particular in bipartite
graphs (grids, trees, hypercubes...), but the computational complexity of
computing remains open in general graphs and even in trees. To progress
further, we propose a notion of monotonicity for the Hunters and Rabbit game
imposing that, roughly, a vertex that has already been shot ``must not host the
rabbit anymore''. This allows us to obtain new results in various graph
classes.
Let the monotone hunter number be denoted by . We show that for any graph with pathwidth , implying
that computing , or even approximating up to an additive
constant, is NP-hard. Then, we show that can be computed in polynomial
time in split graphs, interval graphs, cographs and trees. These results go
through structural characterisations which allow us to relate the monotone
hunter number with the pathwidth in some of these graph classes. In all cases,
this allows us to specify the hunter number or to show that there may be an
arbitrary gap between and , i.e., that monotonicity does not help. In
particular, we show that, for every , there exists a tree with
and . We conclude by proving that computing (resp., )
is FPT parameterised by the minimum size of a vertex cover.Comment: A preliminary version appeared in MFCS 2023. Abstract shortened due
to Arxiv submission requirement
Optimizing the trade-off between number of cops and capture time in Cops and Robbers
The cop throttling number of a graph for the game of Cops and
Robbers is 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 in which both players play optimally. In
this paper, we construct a family of graphs having ,
establish a sublinear upper bound on the cop throttling number, and show that
the cop throttling number of chordal graphs is . We also introduce
the product cop throttling number as a parameter that
minimizes the person-hours used by the cops. This parameter extends the notion
of speed-up that has been studied in the context of parallel processing and
network decontamination. We establish bounds on the product cop throttling
number in terms of the cop throttling number, characterize graphs with low
product cop throttling number, and show that for a chordal graph ,
.Comment: 19 pages, 3 figure
A note on deterministic zombies
"Zombies and Survivor" is a variant of the well-studied game of "Cops and
Robber" where the zombies (cops) can only move closer to the survivor (robber).
We consider the deterministic version of the game where a zombie can choose
their path if multiple options are available. The zombie number, like the cop
number, of a graph is the minimum number of zombies, or cops, required to
capture the survivor. In this short note, we solve a question by Fitzpatrick et
al., proving that the zombie number of the Cartesian product of two graphs is
at most the sum of their zombie numbers. We also give a simple graph family
with cop number and an arbitrarily large zombie number.Comment: 4 page
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