= Solution
For a compact triangulated surface, the <Euler characteristic> is $\chi=V-E+F$, <independent> of the triangulation. For a connected closed orientable surface, the <genus> $g$ is its number of handles, characterized by
$$
\boxed{\chi=2-2g.}
$$
A connected compact orientable surface with $b$ boundary components instead has $\chi=2-2g-b$. For multiple connected components the Euler characteristics add, with <genus> specified for each component. A compact boundaryless surface embedded in $\mathbb R^3$ is orientable, so the usual closed-surface formula applies to each connected component.
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