Menu
An open dominating set for a digraph \(D\) is a subset \(S\) of vertices of \(D\) such that every vertex of \(D\) is adjacent from some vertex of \(S\). The cardinality of a minimum open dominating set for \(D\) is the open domination number \(\rho_1(D)\) of \(D\). The lower orientable open domination number \(\text{dom}_1(G)\) of a graph \(G\) is the minimum open domination number among all orientations of \(G\). Similarly, the upper orientable open domination number \(\text{DOM}_1(G)\) of \(G\) is the maximum such open
domination number. For a connected graph \(G\), it is shown that \(\text{dom}_1(G)\) and \(\text{DOM}_1(G)\) exist if and only if \(G\) is not a tree. A discussion of the upper orientable open domination number of complete graphs is given. It is shown that for each integer \(c\) with \(\text{dom}_1(K_n) \leq c \leq \text{DOM}_1(K_n)\), there exists an
orientation \(D\) of \(K_n\) such that \(\rho_1(D) = c\).