2-DOMINATION Polynomials of Graphs

arXiv: Combinatorics, 2017

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G,x)$ is called a total domination root of $G$. An irrelevant edge of $D_t(G,x)$ is an edge $e \in E$, such that $D_t(G, x) = D_t(G\setminus e, x)$. In this paper, we characterize edges possessing this property. Also we obtain some results for the number of total dominating sets of a regular graph. Finally, we study graphs with exactly two total domination roots $\{-3,0\}$, $\{-2,0\}$ and $\{-1,0\}$.

2010

The domination polynomial of a graph G of order n is the polynomial D(G, x) = P n i=γ(G) d(G, i)x i , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper, we obtain some properties of the coefficients of D(G, x). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by G ′ (m), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs G ′ (m), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if n ≡ 0, 2(mod 3) and D(G, x) = D(Cn, x), then G = Cn.

2014

Let G = (V,E) be a simple graph. The domination polynomial of G is the polynomial D(G, x) = n i=0 d(G, i)xi, where d(G, i) is the number of dominating sets of G of size i. In this paper, we present some new approaches for computation of domination polynomial of specific graphs.

Opuscula Mathematica, 2010

The domination polynomial of a graph G of order n is the polynomial D(G, x) = P n i=γ(G) d(G, i)x i , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper, we obtain some properties of the coefficients of D(G, x). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by G ′ (m), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs G ′ (m), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if n ≡ 0, 2(mod 3) and D(G, x) = D(Cn, x), then G = Cn.

atlas-conferences.com

Let G be a simple graph of order n, the vertex domination polynomial of G is the polynomial D 0 (G, x) = n i=γ 0 (G) d 0 (G, i)x i , where d 0 (G, i) is the number of vertex dominating sets of G with size i, and γ 0 (G) is the vertex domination number of G. Similarly, the edge domination polynomial of G is the polynomial D 1 (G, x) = |E(G)| i=γ 1 (G) d 1 (G, i)x i , where d 1 (G, i) is the number of edge dominating sets of G with size i, and γ 1 (G) is the edge domination number of G. In this paper, we obtain some properties of the coefficients of the edge domination polynomial of G and show that the edge domination polynomial of G is equal to the vertex domination polynomial of line graph L(G) of G.

International Journal for Research in Applied Science and Engineering Technology, 2021

A vertex subset S of a graph G = (V,E) is called a (1,2)-dominating set if S is having the property that for every vertex v in V- S there is atleast one vertex in S of distance 1 from v and a vertex in S at a distance atmost 2 from v. The minimum cardinality of a (1, 2)-dominating set of G, denoted by ϒ (1, 2)(G), is called the (1, 2)-domination number of G. In this paper we discuss about the (1, 2)-dominating set of Shell graph C(n,n-3,), Jewel graph Jn and Comb graph Pn ʘ K1.

arXiv: Combinatorics, 2014

The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$, and $\gamma(G)$ is the domination number of $G$. The roots of domination polynomial is called domination roots. In this article, we compute the domination polynomial and domination roots of all graphs of order less than or equal to 6, and show them in the tables.

2010

a b s t r a c t In this paper, we show that if a graph G has two distinct domination roots, then Z (D(G, x)) = {−2, 0}. Also, if G is a graph with no pendant vertex and has three distinct domination

A domination in graphs is part of graph theory which has many applications. Its application includes the morphological analysis, computer network communication, social network theory, CCTV installation, and many others. A set D of vertices of a simple graph G, that is a graph without loops and multiple edges, is called a dominating set if every vertex u ∈ V (G) − D is adjacent to some vertex v ∈ D. The domination number of a graph G, denoted by γ(G), is the order of a smallest dominating set of G. A dominating set D with |D| = γ(G) is called a minimum dominating set, see Haynes and Henning [5]. This research aims to find the domination number of some families of special graphs, namely Spider Web graph W b n , Helmet graph H n,m , Parachute graph P c n , and any regular graph. The results shows that the resulting domination numbers meet the lower bound of an obtained lower bound γ(G) of any graphs.

2009

We introduce a domination polynomial of a graph G.