The ultracenter and central fringe of a graph

The central distance of a central vertex v in a connected graph G with rad G < diam G is the largest nonnegative integer n such that whenever x is a vertex with d ( v, x ) ≤ n then x is also a central vertex. The subgraph induced by those central vertices of maximum central distance is the ultracenter of G . The subgraph induced by the central vertices having central distance 0 is the central fringe of G . For a given graph G , the smallest order of a connected graph H is determined whose ultracenter is isomorphic to G but whose center is not G . For a given graph F , we determine the smallest order of a connected graph H whose central fringe is isomorphic to G but whose center is not G . © 2001 John Wiley & Sons, Inc.