Abstract
The eccentric digraph ED ( G ) of a digraph G represents the binary relation, defined on the vertex set of G, of being ‘eccentric’; that is, there is an arc from u to v in ED ( G ) if and only if v is at maximum distance from u in G. A digraph G is said to be eccentric if there exists a digraph H such that G = ED ( H ) . This paper is devoted to the study of the following two questions: what digraphs are eccentric and when the relation of being eccentric is symmetric. We present a characterization of eccentric digraphs, which in the undirected case says that a graph G is eccentric iff its complement graph G ¯ is either self-centered of radius two or it is the union of complete graphs. As a consequence, we obtain that all trees except those with diameter 3 are eccentric digraphs. We also determine when ED ( G ) is symmetric in the cases when G is a graph or a digraph that is not strongly connected.
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