scholarly journals Lower bounds for the domination number and the total domination number of direct product graphs

2010 ◽  
Vol 310 (23) ◽  
pp. 3310-3317 ◽  
Author(s):  
Gašper Mekiš
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Product Graphs

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 Dominating the Direct Product of Two Graphs through Total Roman Strategies

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1438 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael G. Yero
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Minimum Degree ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Product Graphs ◽  
Roman Dominating Function ◽  
Dominating Function

Given a graph G without isolated vertices, a total Roman dominating function for G is a function f:V(G)→{0,1,2} such that every vertex u with f(u)=0 is adjacent to a vertex v with f(v)=2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number γtR(G) of G is the smallest possible value of ∑v∈V(G)f(v) among all total Roman dominating functions f. The total Roman domination number of the direct product G×H of the graphs G and H is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between γtR(G×H) and some classical domination parameters for the factors are given. Characterizations of the direct product graphs G×H achieving small values (≤7) for γtR(G×H) are presented, and exact values for γtR(G×H) are deduced, while considering various specific direct product classes.

Lower bounds on the total domination number of a graph

2014 ◽  
Vol 31 (1) ◽  
pp. 52-66 ◽  
Author(s):  
Wyatt J. Desormeaux ◽  
Michael A. Henning
Keyword(s):  
Lower Bounds ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number

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 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.

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