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If the distance between two vertices \(u\) and \(v\) in a graph \(G\) is \(k\), then \(u\) and \(v\) are said to \(k\)-step dominate each other. A set \(S\) of vertices of \(G\) is a \(k\)-step dominating set if every vertex of \(G\) is \(k\)-step dominated by some vertex of \(S\). The minimum cardinality of a \(k\)-step dominating set is the \(k\)-step domination number \(\rho_k(G)\) of \(G\). A sequence \(s: \ell_1, \ell_2, \ldots, \ell_k\) of positive integers is called an orbital dominating sequence for \(G\) if there exist distinct vertices \(v_1, v_2, \ldots, v_k\) of \(G\) such that every vertex of \(G\) is \(\ell_i\)-step dominated by \(v_i\) for some \(i\) (\(1 \leq i \leq k\)). An orbital dominating sequence \(s$ is minimal if no proper subsequence of \(s\) is an orbital dominating sequence for \(G\). The minimum length of a minimal orbital dominating sequence is the orbital domination number \(\gamma_{o}(G)\), while the maximum length of such a sequence is the upper orbital domination number \(\Gamma_{o}(G)\) of \(G\).
It is shown that for every pair \(i, j\) of positive integers with \(i < j\), there exist graphs \(G\) and \(H\) such that both \(\rho_i(G) – \rho_j(G)\) and \(\rho_j(H) – \rho_i(H)\) are arbitrarily large. Also, there exist graphs \(G\) of arbitrarily large radius such that \(\gamma_{o}(G) < \rho_i(G)\) for every integer \(i\) (\(1 \leq i \leq \text{rad} G\)). All trees \(T\) with \(\gamma_{o}(T) = 3\) are characterized, as are all minimum orbital sequences of length 3 for graphs. All graphs \(G\) with \(\Gamma_{o}(G) = 2\) are characterized, as are all trees \(T\) with \(\Gamma_{o}(T) = 3\).