Total domination polynomials of graphs

TL;DR

This paper studies total domination polynomials in graphs, deriving formulas, calculating for specific graph classes, establishing bounds for trees, and exploring relationships with vertex sets.

Contribution

It introduces reduction formulas for total domination polynomials and characterizes graphs attaining bounds, advancing domination theory analysis.

Findings

Derived vertex-reduction and edge-reduction formulas.

Computed total domination polynomials for paths and cycles.

Established sharp upper bounds for trees and characterized extremal graphs.

Abstract

Given a graph $G$, a total dominating set $D_t$ is a vertex set that every vertex of $G$ is adjacent to some vertices of $D_t$ and let $d_t(G,i)$ be the number of all total dominating sets with size $i$. The total domination polynomial, defined as $D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$, recently has been one of the considerable extended research in the field of domination theory. In this paper, we obtain the vertex-reduction and edge-reduction formulas of total domination polynomials. As consequences, we give the total domination polynomials for paths and cycles. Additionally, we determine the sharp upper bounds of total domination polynomials for trees and characterize the corresponding graphs attaining such bounds. Finally, we use the reduction-formulas to investigate the relations between vertex sets and total domination polynomials in $G$.