Graphs G Where $$G-N[v]$$ is a Tree for Each Vertex v

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 \({\mathcal {F}}\) be a family of graphs: to characterize all graphs G such that \(G-N[v]\in {\mathcal {F}}\) for every vertex v of G. Let \({\mathcal {T}}_m\) be the set of all trees of size \(m\ge 0\) for a fixed nonnegative integer m, \({\mathcal {P}}=\{P_t:\ t>0\}\) and \({\mathcal {S}}=\{K_{1,t}:\ t\ge 0\}\). We show that for a connected graph G with its complement \({\overline{G}}\) being connected, \(G-N[v]\in {\mathcal {T}}_m\) for each \(v\in V(G)\) if and only if one of the following holds: \(G-N[v]\cong K_{1,m}\) for each \(v\in V(G)\), or \(G-N[v]\cong P_{m+1}\) for each \(v\in V(G)\). Indeed, the graphs with later two properties are characterized by the same authors very recently (Graphs G in which \(G-N[v]\) has a prescribed property for each vertex v, Discrete Appl. Math., In press.). In addition, we characterize all graphs G such that \(G-N[v]\in {\mathcal {S}}\) for each \(v\in V(G)\) and all graphs G such that \(G-N[v]\in {\mathcal {P}}\) for each \(v\in V(G)\). This solves an open problem raised by Yu and Wu (Graphs in which \(G-N[v]\) is a cycle for each vertex v, Discrete Math. 344 (2021) 112519). Finally, a number of conjectures are proposed for the perspective of the problem.