13,667 research outputs found
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
Edge Roman domination on graphs
An edge Roman dominating function of a graph is a function satisfying the condition that every edge with
is adjacent to some edge with . The edge Roman
domination number of , denoted by , is the minimum weight
of an edge Roman dominating function of .
This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad stating that if is a graph of maximum degree
on vertices, then . While the counterexamples having the edge Roman domination numbers
, we prove that is an upper bound for connected graphs. Furthermore, we
provide an upper bound for the edge Roman domination number of -degenerate
graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic
graphs.
In addition, we prove that the edge Roman domination numbers of planar graphs
on vertices is at most , which confirms a conjecture of
Akbari and Qajar. We also show an upper bound for graphs of girth at least five
that is 2-cell embeddable in surfaces of small genus. Finally, we prove an
upper bound for graphs that do not contain as a subdivision, which
generalizes a result of Akbari and Qajar on outerplanar graphs
On global location-domination in graphs
A dominating set of a graph is called locating-dominating, LD-set for
short, if every vertex not in is uniquely determined by the set of
neighbors of belonging to . Locating-dominating sets of minimum
cardinality are called -codes and the cardinality of an LD-code is the
location-domination number . An LD-set of a graph is global
if it is an LD-set of both and its complement . The global
location-domination number is the minimum cardinality of a
global LD-set of . In this work, we give some relations between
locating-dominating sets and the location-domination number in a graph and its
complement.Comment: 15 pages: 2 tables; 8 figures; 20 reference
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