A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u∈V(D)\S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γd(G), is the maximum directed domination number γ(D) over all orientations D of G. The directed domination number of a complete graph was first studied by Erds [P. Erds On a problem in graph theory, Math. Gaz. 47 (1963) 220222], albeit in a disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if α denotes the independence number of a graph G, we show that α≤Γd(G) ≤α(1+2ln(nα)). © 2011 Elsevier B.V. All rights reserved.
Caro, Y., & Henning, M. A. (2011). A Greedy Partition Lemma for directed domination. Discrete Optimization, 8(3), 452–458. https://doi.org/10.1016/j.disopt.2011.03.003