On orbital domination numbers of graphs (Q2744371)

If the distance between the vertices \(u,v\) of a graph \(G\) is equal to \(k\), then \(u\) and \(v\) are said to \(k\)-step dominate each other. A subset \(S\) of the vertex set \(V(G)\) of \(G\) is called a \(k\)-step dominating set in \(G\), if every vertex of \(G\) is \(k\)-step dominated by a vertex of \(S\) The minimum number of vertices of a \(k\)-step dominating set in \(G\) is the \(k\)-step domination number \(\rho_k(G)\) of \(G\). A sequence \(\ell_1,\ell_2, \dots, \ell_k\) of positive integers is called an orbital sequence for \(G\), if \(G\) contains distinct vertices \(v_1,v_2, \dots,v_k\) such that every vertex of \(G\) is \(\ell_i \)-step dominated by \(v_i\) for some \(i\in\{1,2, \dots,k\}\). The minimum length of an orbital sequence for \(G\) is the orbital domination number \(\gamma_0(G)\) of \(G\). Also the upper orbital domination number \(\Gamma_0(G)\) of \(G\) is defined. The properties of these concepts are studied. A particular attention is paid to the case of trees.