Semicomplete absorbent sets in digraphs

Abstract

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.