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 n is a v ertex with d(v, x) less than or equal to 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 havi ng 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 is omorphic 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 isomorphi c to G but whose center is not G. (C) 2001 John Wiley & Sons, Inc.