ON CONNECTED TOTAL DOMINATING SETS AND CONNECTED TOTAL DOMINATION POLYNOMIAL OF CORONA GRAPHS

2019 ◽  
Vol 22 (1) ◽  
pp. 103-115
Author(s):  
Giovannie M. Entero ◽  
Ariel C. Pedrano
Keyword(s):  
Total Domination ◽  
Dominating Sets ◽  
Domination Polynomial ◽  
Corona Graphs

scholarly journals ON THE ROOTS OF TOTAL DOMINATION POLYNOMIAL OF GRAPHS, II

2019 ◽  
pp. 659
Author(s):  
Saeid Alikhani ◽  
Nasrin Jafari
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination Polynomial

Let $G = (V, E)$ be a simple graph of order $n$. A  total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in  $D$. The total domination number of $G$ is  minimum cardinality of  total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the total domination polynomial of some graphs.  We show that  all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence, we prove that if $\delta\geq \frac{2n}{3}$,  then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.

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.

scholarly journals Perfect dominating sets and perfect domination polynomial of a star graph

2020 ◽  
Vol 8 (4) ◽  
pp. 1751-1755
Author(s):  
Paul Hawkins P. ◽  
Anto A.M. ◽  
Shyla Isac Mary T.
Keyword(s):  
Star Graph ◽  
Dominating Sets ◽  
Domination Polynomial

Strong nonsplit dominating sets and strong nonsplit domination polynomial of complement of paths

2020 ◽  
Vol 12 (06) ◽  
pp. 2050082
Author(s):  
D. Kiruba Packiarani ◽  
Y. Therese Sunitha Mary
Keyword(s):  
Domination Number ◽  
Recursive Formula ◽  
Simple Graph ◽  
Dominating Sets ◽  
The Family ◽  
Domination Polynomial

Let [Formula: see text] be a simple graph of order [Formula: see text]. The strong nonsplit domination polynomial of a graph [Formula: see text] is [Formula: see text] where [Formula: see text] is the number of strong nonsplit dominating sets of [Formula: see text] of size [Formula: see text] and [Formula: see text] is the strong nonsplit domination number of [Formula: see text]. Let [Formula: see text] be the family of strong nonsplit dominating sets of a complement of a path [Formula: see text] ([Formula: see text]) with cardinality [Formula: see text], and let [Formula: see text]. In this paper, we construct [Formula: see text], the recursive formula for [Formula: see text] and [Formula: see text], the strong nonsplit domination polynomial of [Formula: see text]. Also, we obtain some properties of the coefficients of this polynomial.

The Number of 2-dominating Sets, and 2-domination Polynomial of a Graph

2021 ◽  
Vol 42 (4) ◽  
pp. 751-759
Author(s):  
F. Movahedi ◽  
M. H. Akhbari ◽  
S. Alikhani
Keyword(s):  
Dominating Sets ◽  
Domination Polynomial
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