A given graph H is called realizable by a graph G if \(G[N(v)]\cong H\) for every vertex v of G. The Trahtenbrot–Zykov problem says that which graphs are realizable. We consider a problem somewhat opposite in a more general setting. Let t be a positive integer. We characterize all graphs G such that \(G-N[v]\in {\mathcal {F}}\) for every vertex v of G, where \({\mathcal {F}}\) is the set of all t-regular graphs. Indeed, for \(1\le t\le 3\) , the graphs are characterized by the same authors recently. Let \(t\ge 4\) . In this paper, we prove that there is no graph G such that \(G-N[u]\in {\mathcal {F}}\) for each vertex \(u\in V(G)\) if \(G-N[v]\cong H\in {\mathcal {F}}\) for some vertex \(v\in V(G)\) and \(diam(H)\ge 4\) . In addition, we characterize all graphs G such that \(G-N[v]\cong H\) for each \(v\in V(G)\) if H is one of two special graphs whose diameter are 2 and 3, respectively.
