We say that G has the property $\mathcal{P}$ if $G-N[v]$ is a cycle for any vertex $v\in V(G)$, where $N[v]$ is the closed neighborhood of v in G. For an integer $l\in \{3,4,5,6\}$, let ${\mathcal{G}}_l$ be a set of graphs defined as follows:

${\mathcal{G}}_3=\{2K_3\}\cup \{G:\text{both }G$ and $\bar{G}$ are connected, and $\bar{G}$ is a triangle-free cubic graph}, where $\bar{H}$ denotes the complement of H,

${\mathcal{G}}_4=\{\overline{L(H)}:H$ is a connected triangle-free cubic graph}, where $L(H)$ denotes the line graph of H,

${\mathcal{G}}_5=\{\overline{G_{20}}\}$, where $G_{20}$ is the icosahedron,

${\mathcal{G}}_6=\{G(5,2)\}$, where $G(5,2)$ is the Petersen graph. Furthermore, let $\mathcal{G}=\{G_1\vee G_2\vee \cdots \vee G_t:G_i\in {\bigcup }_{l\in \{3,4,5,6\}}{\mathcal{G}}_l$, t is any positive integer}. We show that a graph G has the property $\mathcal{P}$ if and only if $G\in \mathcal{G}$.

我们说G有性质$\mathcal{磷}$ 如果 $G-N[v]$ 是任意顶点的循环 $v\in 伏(G)$， 在哪里 $N[v]$是的封闭附近v在ģ。对于整数$升\in \{3,4,5,6\}$， 让 ${\mathcal{G}}_{升}$ 是一组定义如下的图：

${\mathcal{G}}_3=\{2{钾}_3\}\cup \{G：\text{两个都 }G$ 和 $\bar{G}$ 是连接的，并且 $\bar{G}$ 是无三角形三次图}，其中 $\bar{H}$表示H的补码，

${\mathcal{G}}_4=\{\overline{升(H)}：H$ 是连通无三角形三次图}，其中 $升(H)$表示H的折线图，

${\mathcal{G}}_5=\{\overline{G_{20}}\}$， 在哪里 $G_{20}$ 是二十面体，

${\mathcal{G}}_6=\{G(5,2)\}$， 在哪里 $G(5,2)$是彼得森图。此外，让$\mathcal{G}=\{G_1\vee G_2\vee \cdots \vee G_{吨}：G_{\mathrm{一世}}\in {\bigcup }_{升\in \{3,4,5,6\}}{\mathcal{G}}_{升}$, t是任何正整数}。我们证明图G具有以下性质$\mathcal{磷}$ 当且仅当 $G\in \mathcal{G}$.