 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 byA 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.