Novel ways of enumerating restrained dominating sets of cycles

Abstract

Let G = (V, E) be a graph. A set S ⊆ 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 γr(G), is the smallest cardinality of a restrained dominating set of G. Let G i n be the family of restrained dominating sets of a graph G of order n with cardinality i, and let dr(Gn, i) = |G i n |. The restrained domination polynomial (RDP) of Gn, Dr(Gn, x) is defined as Dr(Gn, x) = n i=γr (Gn ) dr(Gn, i)x i . In this paper, we focus on the RDP of cycles and have, thus, introduced several novel ways to compute dr(Cn, i), where Cn is a cycle of order n. In the first approach, we use a recursive formula for dr(Cn, i); while in the other approach, we construct a generating function to compute dr(Cn, i). We also develop an algorithm, based on integer partitioning and circular permutation, to compute dr(Cn, i). This gives us an upper bound on the number of restrained dominating sets of a fixed size for Cn.