Independent Dominating Polynomial in Graphs

arXiv (Cornell University), 2018

An independent dominating set of the simple graph G = (V, E) is a vertex subset that is both dominating and independent in G. The independent domination polynomial of a graph G is the polynomial D i (G, x) = A x |A| , summed over all independent dominating subsets A ⊆ V. A root of D i (G, x) is called an independence domination root. We investigate the independent domination polynomials of some generalized compound graphs. As consequences, we construct graphs whose independence domination roots are real. Also, we consider some certain graphs and study the number of their independent dominating sets.

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 $\{-1,0\}$.

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.

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.

2009

We introduce a domination polynomial of a graph G.

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.

IOSR Journal of Mathematics, 2016

Let G be a simple connected graph. The connected domination polynomial of G is defined by C d (G,x) =  ) () (G V G i d  c d (G,i) x i , where d  (G) is the minimum cardinality of connected dominating set of G. In this paper, we find the connected dominating polynomial and roots of some general graphs.

International journal of health sciences

Let G be a simple graph of order m. Let D2(G, i) be the family of 2-dominating sets in G with size i. The polynomial D2(G, ) = is called the 2-domination polynomial of G. Let D2(Sm, i) be the family of 2-dominating sets of the spider graph Sm with cardinality i and let d2(Sm, i) = |D2(Sm, i)|. Then the 2-domination polynomial D2(Sm, ) of Sm is defined as D2(Sm, ) = , where is the 2-domination number of Sm. In this paper, we obtain some operations on graphs.

Results in Mathematics, 1990

A set S of vertices of a graph G is dominating if each vertex z not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number,-y(G),

Journal of Discrete Mathematical Sciences and Cryptography, 2016

An edge domination polynomial of a graph G is the polynomial = (,) = (,) , () m t e e t e D G x d G t x G γ ∑ where (,) e d G t is the number of edge dominating sets of G of cardinality t. In this paper, we provide tables which contain coefficient of edge domination polynomial of path and cycle. Also, certain properties of edge dominating polynomial are given.