Semitotal domination number of some graph operations

From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs

Symmetry ◽

10.3390/sym13061036 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1036

Author(s):

Abel Cabrera Martínez ◽

Alejandro Estrada-Moreno ◽

Juan Alberto Rodríguez-Velázquez

Keyword(s):

Vertex Cover ◽

Domination Number ◽

Theoretical Computer Science ◽

Discrete Mathematics ◽

Special Issue ◽

Total Domination Number ◽

Theoretical Computer ◽

Graph Parameters ◽

Semitotal Domination ◽

Domination In Graphs

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard.

Download Full-text

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

International Journal of Engineering and Technology ◽

10.21817/ijet/2016/v8i6/160806234 ◽

2016 ◽

Vol 8 (6) ◽

pp. 2749-2771

Author(s):

Yamuna M ◽

Karthika K

Keyword(s):

Domination Number ◽

Graph Operations

Download Full-text

On the semitotal domination number of line graphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2018.06.010 ◽

2019 ◽

Vol 254 ◽

pp. 295-298 ◽

Cited By ~ 3

Author(s):

Enqiang Zhu ◽

Chanjuan Liu

Keyword(s):

Domination Number ◽

Line Graphs ◽

Semitotal Domination

Download Full-text

Some New Perspectives on Global Domination in Graphs

ISRN Combinatorics ◽

10.1155/2013/201654 ◽

2013 ◽

Vol 2013 ◽

pp. 1-4 ◽

Cited By ~ 4

Author(s):

S. K. Vaidya ◽

R. M. Pandit

Keyword(s):

Dominating Set ◽

Domination Number ◽

Affirmative Answer ◽

Graph Operations ◽

Domination In Graphs ◽

Global Domination Number

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. Here we explore the possibility to relate the domination number of graph G and the global domination number of the larger graph obtained from G by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.

Download Full-text