Domination polynomial of lexicographic product of specific graphs

Advances and Applications in Discrete Mathematics, 2021

A set is a connected total dominating set of the graph G if and only if S admits the following: a dominating set of G, that is for every vertex in is adjacent to at least one vertex in S, equivalently, a total dominating set of G, that is for every vertex in there exists a vertex such that v is adjacent to u, equivalently, and the induced subgraph by the total dominating set S of G which is connected. The connected total domination number of the graph G is the minimum cardinality taken over all connected total dominating sets of G. The connected total domination polynomial of the graph G is defined as where is the order of G, is the connected total domination number of G, and where is the family of connected total dominating sets of G. In this paper, we obtain the connected total domination number of the lexicographic graph the connected total dominating sets of the lexicographic graph and the connected total domination polynomial of the lexicographic graph

2013

The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various products, ranging from explicit formulae and recurrences for specific families to more general results. As an application, we show the domination polynomial is computationally hard to evaluate.

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.

Discret. Appl. Math., 2021

In this paper we begin an exploration of several domination-related parameters (among which are the total, restrained, total restrained, paired, outer connected and total outer connected domination numbers) in the generalized lexicographic product (GLP for short) of graphs. We prove that for each GLP of graphs there exist several equality chains containing these parameters. Some known results on standard lexicographic product of two graphs are generalized or/and extended. We also obtain results on well $\mu$-dominated GLP of graphs, where $\mu$ stands for any of the above mentioned domination parameters. In particular, we present a characterization of well $\mu$-dominated GLP of graphs in the cases when $\mu$ is the domination number or the total domination number.

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.

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.

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.

2000

In this paper, we study the domination number, the global dom ination number, the cographic domination number, the global co graphic domination number and the independent domination number of all the graph products which are non-complete ...