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arXiv 2106.06702math.CO
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再论图的全控制多项式和$\mathcal{D}_t$ -等价类

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

Saeid Alikhani, Nasrin Jafari

备注：
14 pages, With Appendix
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摘要：

设$G = (V, E)$是阶为$n$的简单图. $G$的全控制集是$V$的一个子集$D$，$V$的每个顶点与$D$的某些顶点相邻. $G$的全控制数等于$G$的全控制集的最小基数，记为$γ_t(G)$ . $G$的全控制多项式是多项式$D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)x^i$，其中$d_t(G,i)$是大小为$i$的$G$的全控制集的数目。如果$D_t(G,x)=D_t(H,x)$，则称两个图$G$和$H$是全控制等价的或简单的$\mathcal{D}_t$ -等价的. $G$的等价类，记作$[G]$，是所有图$\mathcal{D}_t$ -等价于$G$的集合。本文研究了某些图的$\mathcal{D}_t$ -等价类。此外，我们还引入了全控制多项式是单峰的图族。

英文摘要：

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 $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)x^i$, 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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