On the period and tail of sequences of iterated eccentric digraphs

The eccentricity e(u) of a vertex u in a digraph G is the maximum distance from u to any other vertex in G. A vertex v is an eccentric vertex of u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G, but with an arc from u to v in ED(G) if and only if v is an eccentric vertex of u in G. Given a positive integer k, the k iterated eccentric digraph of G is defined as ED(G) = ED(ED(G)), where ED(G) = G. In this paper we study the behaviour of the sequence {ED(G)}k when G is a labelled [unlabelled] digraph. More precisely, we investigate the [iso-]period and tail of G defined as the smallest integer numbers p > 0 and t ≥ 0 such that ED(G) = ED(G)