Eccentricity Sequences and Eccentricity Sets in Digraphs

Abstract

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 non-decreasing 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 \(S = \{e(v), v \in V(G)\}\). 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.