245 research outputs found
Strong geodetic problem on Cartesian products of graphs
The strong geodetic problem is a recent variation of the geodetic problem.
For a graph , its strong geodetic number is the cardinality of
a smallest vertex subset , such that each vertex of lies on a fixed
shortest path between a pair of vertices from . In this paper, the strong
geodetic problem is studied on the Cartesian product of graphs. A general upper
bound for is determined, as well as exact values
for , , and certain prisms.
Connections between the strong geodetic number of a graph and its subgraphs are
also discussed.Comment: 18 pages, 9 figure
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
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