24 research outputs found
Maker-Breaker domination game on trees when Staller wins
In the Maker-Breaker domination game played on a graph , Dominator's goal
is to select a dominating set and Staller's goal is to claim a closed
neighborhood of some vertex. We study the cases when Staller can win the game.
If Dominator (resp., Staller) starts the game, then
(resp., ) denotes the minimum number of moves Staller
needs to win. For every positive integer , trees with are characterized. Applying hypergraphs, a general upper bound on
is proved. Let be the subdivided
star obtained from the star with edges by subdividing its edges times, respectively. Then is
determined in all the cases except when and each is even. The
simplest formula is obtained when there are are at least two odd s. If
and are the two smallest such numbers, then . For caterpillars,
exact formulas for and for are
established
How fast can Dominator win in the Maker--Breaker domination game?
We study the Maker--Breaker domination games played by two players, Dominator
and Staller. We give a structural characterization for graphs with
Maker--Breaker domination number equal to the domination number. Specifically,
we show how fast Dominator can win in the game on , for
Maker-Breaker domination number
The Maker-Breaker domination game is played on a graph by Dominator and
Staller. The players alternatively select a vertex of that was not yet
chosen in the course of the game. Dominator wins if at some point the vertices
he has chosen form a dominating set. Staller wins if Dominator cannot form a
dominating set. In this paper we introduce the Maker-Breaker domination number
of as the minimum number of moves of Dominator to
win the game provided that he has a winning strategy and is the first to play.
If Staller plays first, then the corresponding invariant is denoted
. Comparing the two invariants it turns out that they
behave much differently than the related game domination numbers. The invariant
is also compared with the domination number. Using the
Erd\H{o}s-Selfridge Criterion a large class of graphs is found for which
holds. Residual graphs are introduced and
used to bound/determine and .
Using residual graphs, and are
determined for an arbitrary tree. The invariants are also obtained for cycles
and bounded for union of graphs. A list of open problems and directions for
further investigations is given.Comment: 20 pages, 5 figure
Maker-Breaker total domination game
Maker-Breaker total domination game in graphs is introduced as a natural
counterpart to the Maker-Breaker domination game recently studied by Duch\^ene,
Gledel, Parreau, and Renault. Both games are instances of the combinatorial
Maker-Breaker games. The Maker-Breaker total domination game is played on a
graph by two players who alternately take turns choosing vertices of .
The first player, Dominator, selects a vertex in order to totally dominate
while the other player, Staller, forbids a vertex to Dominator in order to
prevent him to reach his goal.
It is shown that there are infinitely many connected cubic graphs in which
Staller wins and that no minimum degree condition is sufficient to guarantee
that Dominator wins when Staller starts the game. An amalgamation lemma is
established and used to determine the outcome of the game played on grids.
Cacti are also classified with respect to the outcome of the game. A connection
between the game and hypergraphs is established. It is proved that the game is
PSPACE-complete on split and bipartite graphs. Several problems and questions
are also posed.Comment: 21 pages, 5 figure
The Maker-Maker domination game in forests
We study the Maker-Maker version of the domination game introduced in 2018 by
Duch\^ene et al. Given a graph, two players alternately claim vertices. The
first player to claim a dominating set of the graph wins. As the Maker-Breaker
version, this game is PSPACE-complete on split and bipartite graphs. Our main
result is a linear time algorithm to solve this game in forests. We also give a
characterization of the cycles where the first player has a winning strategy
Avoider-Enforcer Game is NP-hard
In an Avoider-Enforcer game, we are given a hypergraph. Avoider and Enforcer alternate in claiming an unclaimed vertex, until all the vertices of the hypergraph are claimed. Enforcer wins if Avoider claims all vertices of an edge; Avoider wins otherwise. We show that it is NP-hard to decide if Avoider has a winning strategy
Avoidance Games Are PSPACE-Complete
Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games are studied since the introduction of the game of SIM in 1968, but only few complexity results are known on them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojakovi\'c proved that these games are NP-hard. As these games corresponds to the mis\`ere version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question remained open since then. We prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete, and as a consequence of it that some particular Avoider-Enforcer games also are
Maker-Breaker total domination games on cubic graphs
We study Maker-Breaker total domination game played by two players, Dominator
and Staller on the connected cubic graphs. Staller (playing the role of Maker)
wins if she manages to claim an open neighbourhood of a vertex. Dominator wins
otherwise (i.e. if he can claim a total dominating set of a graph). For certain
graphs on vertices, we give the characterization on those which are
Dominator's win and those which are Staller's win