Perfect Domination Polynomial of Homogeneous Caterpillar Graphs and of Full Binary Trees

Let $G=(V,E)$ be a simple graph of order $n$. A set $S \subseteq V(G)$ is a perfect dominating set of a graph $G$ if every vertex $v\in V(G)-S$ is adjacent to exactly one vertex in $S$. That is, every vertex outside $S$ has exactly one neighbor in $S$. Every graph $G$ has at least the trivial perfect dominating sets consisting of all vertices in $G$. The perfect domination number $\gamma_{pf} (G)$ is the minimal cardinality of dominating sets in $G$. Let $D_{pf} (G,i)$ be the family of perfect dominating sets for a graph $G$ with cardinality $i$ and $d_{pf} (G,i)= |D_{pf} (G,i)|$. The perfect domination polynomial of a graph $G$ of order $n$ is
$$D_{pf} (G,x)=\sum_{i=\gamma_{pf}(G)}^{n} d_{pf}(G,i)x^n,$$
where $d_{pf} (G,i)$ is the number of perfect dominating sets of $G$ of size $i$. In this paper, we studied the perfect domination polynomial $D_{pf} (G,x)$ of homogeneous caterpillar graphs and of full binary trees.