A subset of vertices $S$ of a graph $G$ is a dominating set if every vertex in $V \setminus S$ has at least one neighbor in $S$. A domatic partition is a partition of the vertices of a graph $G$ into disjoint dominating sets. The domatic number $d(G)$ is the maximum size of a domatic partition. Suppose that $dp(G,i)$ is the number of distinct domatic partition of $G$ with cardinality $i$. In this paper, we consider the generating function of $dp(G,i)$, i.e., $DP(G,x)=\sum_{i=1}^{d(G)}dp(G,i)x^i$ which we call it the domatic partition polynomial. We explore the domatic polynomial for trees, providing a quadratic time algorithm for its computation based on weak 2-coloring numbers. Our results include specific findings for paths and certain graph products, demonstrating practical applications of our theoretical framework.

翻译：图$G$的顶点子集$S$若满足$V \setminus S$中每个顶点至少与$S$中一个顶点相邻，则称$S$为支配集。支配划分是将图$G$的顶点划分为互不相交的支配集。支配数$d(G)$是支配划分的最大基数。设$dp(G,i)$表示基数为$i$的不同支配划分的数量。本文研究$dp(G,i)$的生成函数$DP(G,x)=\sum_{i=1}^{d(G)}dp(G,i)x^i$，称其为支配划分多项式。我们探究了树的支配多项式，基于弱2着色数提出了二次时间计算算法。研究结果包含对路径及特定图积的具体结论，展示了理论框架的实际应用价值。