Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set 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 denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...

let $g$ be a simple graph of order $n$‎. ‎we consider the‎ ‎independence polynomial and the domination polynomial of a graph‎ ‎$g$‎. ‎the value of a graph polynomial at a specific point can give‎ ‎sometimes a very surprising information about the structure of the‎ ‎graph‎. ‎in this paper we investigate independence and domination‎ ‎polynomial at $-1$ and $1$‎.

A graph $G$ is called $P_4$-free, if $G$ does not contain an induced subgraph $P_4$. The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,x)$ is called a domination root of $G$. In this paper we state and prove formula for the domination polynomial of no...

let $g$ be a simple graph of order $n$. the domination polynomial of $g$ 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 $g$. in this paper we present some families of graphs whose domination polynomials are unimodal.

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ 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 $G$. In this paper we present some families of graphs whose domination polynomials are unimodal.

We consider two graph optimization problems called vector domination and total vector domination. In vector domination one seeks a small subset S of vertices of a graph such that any vertex outside S has a prescribed number of neighbors in S. In total domination, the requirement is extended to all vertices of the graph. We prove that these problems cannot be approximated to within a factor of c...

The domination polynomial of a graph G of order n is the polynomial D(G, x) = Pn i=γ(G) d(G, i)x , 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 rela...