TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Yazd University, 89195-741, Yazd, Iran.

2 Department of Mathematics, Yazd University, 89195-741 Yazd, Iran.

Abstract

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 number of total dominating sets of $G$ of size $i$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two
vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$.
In this paper, we consider some particular cases of these graphs that most of them are of importance in chemistry and study their total domination polynomials.

Keywords

Journal of Algebraic Systems
Volume 5, Issue 2
January 2018
Pages 127-138

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