Graphs In Which G-N[v]Induces A Cycle Foreach Vertex V

Let G be a graph with vertex set V(G)and edge set E(G).We say that G has the property P if G-N[v]is a cycle for any vertex v ∈ V(G),where N[v]is the closed neighborhood of v in G.Our main results are summarized as follows:in the first part,we characterize all graphs that satisfy G-N[v](?)Cl(l ∈ ｛3,4,5,6}).They are characterized as follows:{G:G-N[v](?)C3｝=｛2K3}∪{G:both G and G are connected,and G is a triangle-free cubic graph};{G:G-N[v](?)C4}={L(H):H is a connected triangle-free cubic graph},where L(H)is the line graph of H;{G:G-N[v](?)C5}={G20};{G:G-N[v](?)C6}={G(5,2)},where G(5,2)is the Petersen graph.In the second part,we prove that for an integer l≥7,there exists no graph G such that G-N[v](?)<Cl for any vertex v ∈ V(G).In the third part,we characterize some special graphs.Both of them and their complements have the property P.Let G be a graph.Then both G and G have the property P if and only if {G,G}={G(5,2),G(5,2)} or {G,G}={L(K3,3),L(K3,3)}.