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Let D = (V, A) be a digraph and S a partition of V (D). We say that S is a strong in-domatic partition if every S in S holds that every vertex not in S has at least one out-neighbor in S, that is S is an in-dominating set, and D S is strongly connected. The maximum number of elements in a strong in-domatic partition is called the strong in-domatic number of D and it is denoted by d − s (D). In this paper we introduce those concepts and determine the value of d − s for semicomplete digraphs and planar digraphs. We show some structural properties of digraphs which have a strong in-domatic partition and we see some bounds for d − s (D). Then we study this concept in the Cartesian product, composition, line digraph and other associated digraphs.In addition, we characterize strong in-domatic critical digraphs and we give two families strong in-domatic critical digraphs which hold some properties, where a strong in-domatic critical digraph D holds that d − s (D − e) = d − s (D) − 1 for every e in A(D).

Let D ¼ ðVðDÞ, AðDÞÞ be a digraph and S a subset of vertices of D, S is an absorbent set if for every v in VðDÞ n S there exists a vertex u in S such that ðv, uÞ 2 AðDÞ: A subset S of V(D) is a semicomplete absorbent set if S is absorbent and the induced subdigraph DhSi is semicomplete. The minimum (respectively maximum) of the cardinalities of the semicomplete absorbent sets is the lower (respectively upper) semicomplete absorbent number, denoted by c sas ðDÞ (respectively C sas ðDÞ). In this paper we introduce the concept of semicomplete absorbent set; we will show some structural properties on the digraphs which have a semicomplete absorbent set and we will present some bounds for c sas ðDÞ and C sas ðDÞ: Then we will study the Cartesian product, the composition of digraphs and the line digraph in relation with those numbers.