On Exact \(n\)-Step Domination

A graph \(G\) with vertex set \(V(G)\) is an exact \(n\)-step domination graph if there is some subset \(S \subseteq V(G)\) such that each vertex in \(G\) is distance \(t\) from exactly one vertex in \(S\). Given a set \(A \subseteq \mathbb{N}\), we characterize cycles \(C_t\) with sets \(S \subseteq V(C_t)\) that are simultaneously \(a\)-step dominating for precisely those \(a \in A\). Using Polya’s method, we compute the number of \(t\)-step dominating sets for a cycle \(C_t\) that are distinct up to automorphisms of \(C_t\). Finally, we generalize the notion of exact \(t\)-step domination.