Novel ways of enumerating restrained dominating sets of cycles

TL;DR

This paper introduces new recursive and generating function methods to compute the restrained domination polynomial of cycles, addressing the NP-hard problem of finding restrained domination numbers.

Contribution

The paper presents novel recursive formulas and generating function techniques specifically for calculating the restrained domination polynomial of cycle graphs.

Findings

Derived recursive formulas for $d_r(C_n,i)$

Constructed generating functions for $d_r(C_n,i)$

Provided efficient methods for computing restrained domination polynomials

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 $d_r(C_n,i)$.