Novel ways of enumerating restrained dominating sets of cycles
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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 γ_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) = ∑_i=γ_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.
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