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\begin{document}
Fix a plane drawing $\widetilde{G}$ of $G$. For simplicity we write $G$ for $\widetilde{G}$.
If $|G|=3$\textcolor{brown}{{} }then \thmref{main result} is trivial. So assume that
$|G|\ge4$. We inductively construct an isometric-path decomposition $P_{1},\ldots,P_{k},k\ge2$
such that:\smallskip

\noindent\begin{minipage}[t][1\totalheight][c]{0.1\columnwidth}%
({*})%
\end{minipage}%
\begin{minipage}[t]{0.85\textwidth}%
for any component $C$ of $G-\bigcup_{1\le j\le k}P_{j}$ the boundary of the region
of $R^{2}\smallsetminus G[P_{1}\cup\ldots\cup P_{k}]$\textcolor{brown}{{} }containing
$C$ is a cycle in $G$ that has its vertices in exactly two paths from $P_{1},\ldots,P_{k}$.%
\end{minipage}
\end{document}
