Step Domination in Graphs

A set \(S = \{v_1, v_2, \ldots, v_n\}\) of vertices in a graph \(G\) with associated sequence \(k_1, k_2, \ldots, k_n\) of nonnegative integers is called a step domination set if every vertex of \(G\) is at distance \(k_i\) from \(v_i\) for exactly one \(i\) (\(1 \leq i \leq n\)). The minimum cardinality of a step domination set is called the step domination number of \(G\). This parameter is determined for several classes of graphs and is investigated for trees.