5,092 research outputs found
3-Factor-criticality in double domination edge critical graphs
A vertex subset of a graph is a double dominating set of if
for each vertex of , where is the set of the
vertex and vertices adjacent to . The double domination number of ,
denoted by , is the cardinality of a smallest double
dominating set of . A graph is said to be double domination edge
critical if for any edge . A double domination edge critical graph with is called --critical. A graph is
-factor-critical if has a perfect matching for each set of
vertices in . In this paper we show that is 3-factor-critical if is
a 3-connected claw-free --critical graph of odd order
with minimum degree at least 4 except a family of graphs.Comment: 14 page
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
Domination parameters with number 2: Interrelations and algorithmic consequences
In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γw2(G), the 2-domination number, γ2(G), the {2}-domination number, γ{2}(G), the double domination number, γ×2(G), the total {2}-domination number, γt{2}(G), and the total double domination number, γt×2(G), where G is a graph in which the corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G), and two classical parameters, the domination number, γ(G), and the total domination number, γt(G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Brešar, Boštjan. Institute of Mathematics, Physics and Mechanics; Eslovenia. University of Maribor; EsloveniaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Milanič, Martin. University of Primorska; EsloveniaFil: Safe, Martin Dario. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin
Domination parameters with number 2: interrelations and algorithmic consequences
In this paper, we study the most basic domination invariants in graphs, in
which number 2 is intrinsic part of their definitions. We classify them upon
three criteria, two of which give the following previously studied invariants:
the weak -domination number, , the -domination number,
, the -domination number, , the double
domination number, , the total -domination number,
, and the total double domination number, , where is a graph in which a corresponding invariant is well
defined. The third criterion yields rainbow versions of the mentioned six
parameters, one of which has already been well studied, and three other give
new interesting parameters. Together with a special, extensively studied Roman
domination, , and two classical parameters, the domination number,
, and the total domination number, , we consider 13
domination invariants in graphs . In the main result of the paper we present
sharp upper and lower bounds of each of the invariants in terms of every other
invariant, large majority of which are new results proven in this paper. As a
consequence of the main theorem we obtain some complexity results for the
studied invariants, in particular regarding the existence of approximation
algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure
Computational complexity aspects of super domination
Let G be a graph. A dominating set D ⊆ V (G) is a super dominating set if for every vertex x ∈ V (G) \ D there exists y ∈ D such that NG (y) ∩ (V (G) \ D)) = {x}. The cardinality of a smallest super dominating set of G is the super domination number of G. An exact formula for the super domination number of a tree T is obtained, and it is demonstrated that a smallest super dominating set of T can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph G is at most a given integer if G is a bipartite graph of girth at least 8. The super domination number is determined for all k-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determinepublishedVersio
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