Menu
Let \(G\) be a graph. The weak domination number of \(G\), \(\gamma_w(G)\), is the minimum cardinality of a set \(D\) of vertices where every vertex \(u \notin D\) is adjacent to a vertex \(v \in D\), where \(\deg(v) \leq \deg(u)\). The strong domination number of \(G\), \(\gamma_s(G)\), is the minimum cardinality of a set \(D\) of vertices where every vertex \(u \notin D\) is adjacent to a vertex \(v \in D\), where \(\deg(v) \geq \deg(u)\). Similarly, the independent weak domination number, \(i_w(G)\), and the independent strong domination number, \(i_{st}(G)\), are defined with the additional requirement that the set \(D\) is independent. We find upper bounds on the number of edges of a graph in terms of the number of vertices and for each of these four domination parameters. We also characterize all graphs where equality is achieved in each of the four bounds.