Eccentric graphs

The eccentricity e(nu) Of a vertex nu in a connected graph G is the distance between nu and a vertex farthest from nu. A vertex is an eccentric vertex of v nu if d(u, nu) = e(nu). A vertex w of G is an eccentric vertex of G if w is an eccentric vertex of some vertex of G, 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 less than or equal to c less than or equal to 2a there exists an eccentric graph G with rad G = a and diam G = 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. (C) 1999 John Wiley & Sons, Inc. Networks 34: 115-121, 1999.