The ultracenter and central fringe of a graph (Q2747803)

Let \(G\) be a connected graph with \(\text{rad}(G)< \text{diam}(G)\) and let \(v\) be a central vertex. The central distance of \(v\) is defined as the largest nonnegative integer \(n\) such that whenever \(d(v,x)\leq n\) the vertex \(x\) is in the center of \(G\). The subgraph induced by those central vertices of maximum central distance is said to be the ultracenter of \(G\). The subgraph induced by the central vertices having central distance 0 is said to be the central fringe of \(G\). In this paper it is shown that (i) for a given graph \(G\), the smallest order of a connected graph \(H\) whose ultracenter is isomorphic to \(G\) but whose center is not \(G\) equals \(|V(G)|+4\); and (ii) for a given graph \(F\), the smallest order of a connected graph \(H\) whose central fringe is isomorphic to \(G\) but whose center is not \(G\) is equal to \(|V(F)|+3\).