Novel ways of enumerating restrained dominating sets of cycles

Let $G = (V, E)$ be a graph. A set $S \subseteq V$ is a restrained dominating set (RDS) if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V - S$. The restrained domination number of $G$, denoted by $\gamma_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. Let $G_n^i$ be the family of restrained dominating sets of a graph $G$ of order $n$ with cardinality $i$, and let $d_r(G_n, i)=|G_n^i|$. The restrained domination polynomial (RDP) of $G_n$, $D_r(G_n, x)$ is defined as $D_r(G_n, x) = \sum_{i=\gamma_r(G_n)}^{n} d_r(G_n,i)x^i$. In this paper, we focus on the RDP of cycles and have, thus, introduced several novel ways to compute $d_r(C_n, i)$, where $C_n$ is a cycle of order $n$. In the first approach, we use a recursive formula for $d_r(C_n,i)$; while in the other approach, we construct a generating function to compute $d_r(C_n,i)$. We also develop an algorithm, based on integer partitioning and circular permutation, to compute $d_r(C_n,i)$. This gives us an upper bound on the number of restrained dominating sets of a fixed size for $C_n$.