Domination in undirected graphs is a well-studied part of graph theory because it has lot of practical applications. The concept of domination in undirected graphs has naturally extended to digraphs. By contrast, convex domination in digraphs have not yet gained the same amount of attention, although it has several useful applications as well. In this paper, we define weakly and strongly convex domination in digraphs and obtaining some results in standard digraphs. A subset S+ of V of a digraph D is said to be weakly convex dominating set if S + is weakly convex and dominating set. The weakly convex domination number of D is the smallest cardinality of a weakly convex dominating set of D and it is denoted by γ_(+wcond)(D). A subset S+ of V of a digraph D is said to be strongly convex dominating set if S is strongly convex and dominating set. The strongly convex domination number of D is the smallest cardinality of a strongly convex dominating set of D and it is denoted by γ_(+scond)(D).
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