Total domination polynomial of graphs from primary subgraphs

TL;DR

This paper investigates the total domination polynomial of graphs constructed from primary subgraphs, especially those relevant in chemistry, providing new formulas and insights into their combinatorial properties.

Contribution

It introduces formulas for the total domination polynomial of graphs built from primary subgraphs via point-attaching, with applications to chemically significant graph structures.

Findings

Derived explicit formulas for total domination polynomials of specific graph classes.

Connected total domination polynomials to chemical graph structures.

Provided computational methods for total domination polynomials in complex graphs.

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.