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Eccentric graphs

✍ Scribed by Chartrand, Gary; Gu, Weizhen; Schultz, Michelle; Winters, Steven J.

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 212 KB
Volume: 34
Category: Article
ISSN: 0028-3045
DOI: 10.1002/(sici)1097-0037(199909)34:2<115::aid-net4>3.0.co;2-k

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✦ Synopsis

The eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex farthest from v.

The eccentricity e(G) of G is the minimum integer k such that every vertex of G with eccentricity at least k is an eccentric vertex. A graph G is an eccentric graph if every vertex of G is an eccentric vertex or, equivalently, if the radius of G equals e(G). It is shown that for every pair a, c of positive integers satisfying a Յ c Յ 2a there exists an eccentric graph G with rad G ϭ a and diam G ϭ c. Moreover, for every connected graph G, there exists a connected graph H containing G as an induced subgraph such that V(G) is the set of eccentric vertices of H if and only if every vertex of G has eccentricity 1 or no vertex of G has eccentricity 1. Similar characterizations are presented for graphs that are the center or periphery of some eccentric graph.

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