scholarly journals Secure Total Domination in Rooted Product Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 600 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Alejandro Estrada-Moreno ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Domination Number ◽  
Factor Graphs ◽  
Total Domination ◽  
Total Domination Number ◽  
Product Graphs

In this article, we obtain general bounds and closed formulas for the secure total domination number of rooted product graphs. The results are expressed in terms of parameters of the factor graphs involved in the rooted product.

scholarly journals On Grundy total domination number in product graphs

2021 ◽  
Vol 41 (1) ◽  
pp. 225 ◽  
Author(s):  
Boštjan Brešar ◽  
Csilla Bujtás ◽  
Tanja Gologranc ◽  
Sandi Klavžar ◽  
Gašper Košmrlj ◽  
...  
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Product Graphs

scholarly journals Total Domination in Rooted Product Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1929
Author(s):  
Abel Cabrera Martínez ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Graph Theory ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Product Graph ◽  
Product Graphs

During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions.

The minus total k-domination numbers in graphs

2021 ◽  
pp. 2150150
Author(s):  
Jonecis Dayap ◽  
Nasrin Dehgardi ◽  
Leila Asgharsharghi ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Sharp Bounds ◽  
Number Formula ◽  
Domination In Graphs ◽  
Dominating Function ◽  
Domination Numbers

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].

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

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.

scholarly journals New Bounds on the Double Total Domination Number of Graphs

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.

Effect of vertex-removal on game total domination numbers

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.

On domination and total domination number of a collaboration graph

2017 ◽  
Vol 9 (5/6) ◽  
pp. 541
Author(s):  
Valentina E. Balas ◽  
V. Yegnanarayanan ◽  
A. Renuka Lakshmi
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number
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