Eccentricity sequences and eccentricity sets in digraphs

The eccentricity e(v) of a vertex v in a strongly connected digraph G is the maximum distance from v. The eccentricity sequence of a digraph is the list of eccentricities of its vertices given in nondecreasing order. A sequence of positive integers is a digraphical eccentric sequence if it is the eccentricity sequence of some digraph. A set of positive integers S is a digraphical eccentric set if there is a digraph G such that S = {e(v), v ∈ V (G)}. In this paper, we present some necessary and sufficient conditions for a sequence S to be a digraphical eccentric sequence. In some particular cases, where either the minimum or the maximum value of S is fixed, a characterization is derived. We also characterize digraphical eccentric sets.