Novel Ways of Enumerating Restrained Dominating Sets of Cycles

Abstract

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. Finding the restrained domination number is NP-hard for bipartite and chordal graphs. 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 $d_r(C_n,i)$.