Fair total  domination number  in cactus graphs

For any integer [Formula: see text], a minus total [Formula: see text]-dominating function is a function [Formula: see text] satisfying [Formula: see text] for every [Formula: see text], where [Formula: see text]. The minimum of the values of [Formula: see text], taken over all minus total [Formula: see text]-dominating functions [Formula: see text], is called the minus total [Formula: see text]-domination number and is denoted by [Formula: see text]. In this paper, we initiate the study of minus total [Formula: see text]-domination in graphs, and we present different sharp bounds on [Formula: see text]. In addition, we determine the minus total [Formula: see text]-domination number of some classes of graphs. Some of our results are extensions of known properties of the minus total domination number [Formula: see text].

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A note on improved upper bounds on the transversal number of hypergraphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500046 ◽

2019 ◽

Vol 11 (01) ◽

pp. 1950004

Author(s):

Michael A. Henning ◽

Nader Jafari Rad

Keyword(s):

Upper Bound ◽

Dominating Set ◽

Domination Number ◽

Minimum Size ◽

Total Domination ◽

Total Domination Number ◽

Minimum Cardinality ◽

Total Dominating Set ◽

Isolated Vertex ◽

Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

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Essential upper bounds on the total domination number

Discrete Applied Mathematics ◽

10.1016/j.dam.2018.03.008 ◽

2018 ◽

Vol 244 ◽

pp. 103-115 ◽

Cited By ~ 1

Author(s):

Michael A. Henning

Keyword(s):

Domination Number ◽

Upper Bounds ◽

Total Domination ◽

Total Domination Number

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New Bounds on the Double Total Domination Number of Graphs

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-021-01200-0 ◽

2021 ◽

Author(s):

A. Cabrera-Martínez ◽

F. A. Hernández-Mira

Keyword(s):

Minimum Degree ◽

Dominating Set ◽

Domination Number ◽

Total Domination ◽

Dominating Sets ◽

Total Domination Number ◽

Minimum Cardinality ◽

Total Dominating Set

AbstractLet G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

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Effect of vertex-removal on game total domination numbers

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501296 ◽

2019 ◽

Vol 13 (07) ◽

pp. 2050129

Author(s):

Karnchana Charoensitthichai ◽

Chalermpong Worawannotai

Keyword(s):

Domination Number ◽

Total Domination ◽

Total Domination Number ◽

Connected Graphs ◽

Isolated Vertex ◽

Vertex Removal ◽

Total Domination Game ◽

Domination Game ◽

Domination Numbers

The total domination game is played on a graph [Formula: see text] by two players, named Dominator and Staller. They alternately select vertices of [Formula: see text]; each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. In this paper, we show that for any graph [Formula: see text] and a vertex [Formula: see text], where [Formula: see text] has no isolated vertex, we have [Formula: see text] and [Formula: see text]. Moreover, all such differences can be realized by some connected graphs.

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