Euler characteristic and holes ($\chi =b_0-b_1+b_2-\cdots$)

For a finite CW complex, $b_0$ counts connected components: its connected components are its path components, as recorded by the zeroth homology group. For a finite graph regarded as a one-dimensional CW complex, no higher homology groups occur, so $b_1=E-V+b_0$ counts independent graph cycles, and $\chi =b_0-b_1$. For a connected planar surface obtained by removing $h$ disjoint open disks from the interior of a closed disk, $(b_0,b_1,b_2)=(1,h,0)$ and $\chi =1-h$; the annulus is $h=1$.

The word "hole" needs its dimension. A two-sphere encloses a cavity but has $b_1=0$ and $b_2=1$. A closed orientable surface of genus $g$ instead has $(b_0,b_1,b_2)=(1,2g,1)$ and $\chi =2-2g$: increasing the genus by one adds two independent one-dimensional homology classes. In higher dimensions, further homology groups enter with alternating signs; there is no universal "components minus handles" rule.