Elsevier

Discrete Mathematics

Volume 306, Issue 2, 6 February 2006, Pages 210-219
Discrete Mathematics

Characterization of eccentric digraphs

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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.

Keywords

Eccentricity
Eccentric vertex
Distance
Eccentric digraph

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1

Supported by the Ministry of Science and Technology, Spain, and the European Regional Development Fund (ERDF) under project TIC-2001-2171.