On graph approximations of surfaces with small area

It is shown that, for every closed oriented surface $M$ of genus $g$ with an arbitrary Riemann metric, there exists a metric graph of genus at most $g$ such that the Gromov-Hausdorff distance between $M$ and $\Gamma$ does not exceed $C\sqrt{{\mathrm{Vol}}M}$, where $C$ depends only on $g$.