  A graph $\textit{G}$ is a tuple $(\textit{V}, \textit{E})$, where
$\textit{V}$ is the vertex set, $\textit{E}$ is the edge set. A reduced graph
is a graph of deleting non-Hamiltonian edges and smoothing out the redundant
vertices of degree 2 on an edge except for leaving only one vertex of degree 2.
We denote by I a set of cycles only jointed by inside vertices. |I| is the
number of sets I in a graph. We use a norm graph to denote a reduced graph of
|I|=1. $\textit{g}$ is a subgraph obtained by deleting all removable cycles
from a basis of a norm graph. In this paper, we show that a norm graph
$\textit{G}$ is non-Hamiltonian, if and only if, $\textit{g}$ and K_2_,_3 are
homeomorphic.
