More on total domination polynomial and ${\mathcal{D}}_t$-equivalence classes of some graphs

Abstract

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 $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply ${\mathcal{D}}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs ${\mathcal{D}}_t$-equivalent to $G$. In this paper, we investigate ${\mathcal{D}}_t$-equivalence classes of some graphs. Also we introduce some families of graphs whose total domination polynomials are unimodal.

arXiv comment14 pages, With Appendix