18,812 research outputs found
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
Game Total Domination Critical Graphs
In the total domination game played on a graph , players Dominator and
Staller alternately select vertices of , as long as possible, such that each
vertex chosen increases the number of vertices totally dominated. Dominator
(Staller) wishes to minimize (maximize) the number of vertices selected. The
game total domination number, , of is the number of
vertices chosen when Dominator starts the game and both players play optimally.
If a vertex of is declared to be already totally dominated, then we
denote this graph by . In this paper the total domination game critical
graphs are introduced as the graphs for which holds for every vertex in . If , then is called --critical. It is proved that the
cycle is -critical if and only if and that the path is -critical if and only
if . --critical and --critical graphs are also characterized as well as --critical joins of graphs
Cutting Lemma and Union Lemma for the Domination Game
Two new techniques are introduced into the theory of the domination game. The
cutting lemma bounds the game domination number of a partially dominated graph
with the game domination number of suitably modified partially dominated graph.
The union lemma bounds the S-game domination number of a disjoint union of
paths using appropriate weighting functions. Using these tools a conjecture
asserting that the so-called three legged spiders are game domination critical
graphs is proved. An extended cutting lemma is also derived and all game
domination critical trees on 18, 19, and 20 vertices are listed
On the game total domination number
The total domination game is a two-person competitive optimization game,
where the players, Dominator and Staller, alternately select vertices of an
isolate-free graph . Each vertex chosen must strictly increase the number of
vertices totally dominated. This process eventually produces a total dominating
set of . Dominator wishes to minimize the number of vertices chosen in the
game, while Staller wishes to maximize it. The game total domination number of
, , is the number of vertices chosen when Dominator
starts the game and both players play optimally.
Recently, Henning, Klav\v{z}ar, and Rall proved that holds for every graph which is given on vertices such
that every component of it is of order at least ; they also conjectured that
the sharp upper bound would be . Here, we prove that
holds for every which contains no
isolated vertices or isolated edges.Comment: 11 page
The variety of domination games
Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total
domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well
established games played on graphs by two players, named Dominator and Staller.
In this paper, Z-domination game, L-domination game, and LL-domination game are
introduced as natural companions of the standard domination games.
Versions of the Continuation Principle are proved for the new games. It is
proved that in each of these games the outcome of the game, which is a
corresponding graph invariant, differs by at most one depending whether
Dominator or Staller starts the game. The hierarchy of the five domination
games is established. The invariants are also bounded with respect to the
(total) domination number and to the order of a graph. Values of the three new
invariants are determined for paths up to a small constant independent from the
length of a path. Several open problems and a conjecture are listed. The latter
asserts that the L-domination game number is not greater than of the
order of a graph.Comment: 21 page
Scaling algorithms for approximate and exact maximum weight matching
The {\em maximum cardinality} and {\em maximum weight matching} problems can
be solved in time , a bound that has resisted improvement
despite decades of research. (Here and are the number of edges and
vertices.) In this article we demonstrate that this " barrier" is
extremely fragile, in the following sense. For any , we give an
algorithm that computes a -approximate maximum weight matching in
time, that is, optimal {\em linear time}
for any fixed . Our algorithm is dramatically simpler than the best
exact maximum weight matching algorithms on general graphs and should be
appealing in all applications that can tolerate a negligible relative error.
Our second contribution is a new {\em exact} maximum weight matching
algorithm for integer-weighted bipartite graphs that runs in time
. This improves on the -time and
-time algorithms known since the mid 1980s, for . Here is the maximum integer edge weight
Complexity of the Game Domination Problem
The game domination number is a graph invariant that arises from a game,
which is related to graph domination in a similar way as the game chromatic
number is related to graph coloring. In this paper we show that verifying
whether the game domination number of a graph is bounded by a given integer is
PSPACE-complete. This contrasts the situation of the game coloring problem
whose complexity is still unknown.Comment: 14 pages, 3 figure
Progress Towards the Total Domination Game -Conjecture
In this paper, we continue the study of the total domination game in graphs
introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players
Dominator and Staller alternately select vertices of . Each vertex chosen
must strictly increase the number of vertices totally dominated, where a vertex
totally dominates another vertex if they are neighbors. This process eventually
produces a total dominating set of in which every vertex is totally
dominated by a vertex in . Dominator wishes to minimize the number of
vertices chosen, while Staller wishes to maximize it. The game total domination
number, , of is the number of vertices chosen when
Dominator starts the game and both players play optimally. Henning, Klav\v{z}ar
and Rall [Combinatorica, to appear] posted the -Game Total
Domination Conjecture that states that if is a graph on vertices in
which every component contains at least three vertices, then . In this paper, we prove this conjecture over the
class of graphs that satisfy both the condition that the degree sum of
adjacent vertices in is at least and the condition that no two vertices
of degree are at distance apart in . In particular, we prove that by
adopting a greedy strategy, Dominator can complete the total domination game
played in a graph with minimum degree at least in at most moves.Comment: 14 page
Domination game and minimal edge cuts
In this paper a relationship is established between the domination game and
minimal edge cuts. It is proved that the game domination number of a connected
graph can be bounded above in terms of the size of minimal edge cuts. In
particular, if a minimum edge cut of a connected graph , then
. Double-Staller graphs
are introduced in order to show that this upper bound can be attained for
graphs with a bridge. The obtained results are used to extend the family of
known traceable graphs whose game domination numbers are at most one-half their
order. Along the way two technical lemmas, which seem to be generally
applicable for the study of the domination game, are proved.Comment: 15 pages, 1 figur
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
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