Can the Euler characteristic be calculated for an open surface?

Question

I have an open, three-dimensional mesh that looks similar to the one drawn here (in that it contains no volume).

I am using ORS Dragonfly, an image processing software, to visualize this mesh and have access to the number of faces, edges, and vertices. Is it possible to calculate the Euler characteristic of this mesh given that it is not a closed surface? And if so, how?

Answer 1

Let $S$ be your mesh, and let $\overline{S}$ be $S$ with its boundary circle $\partial S$ added it. Because Euler characteristic plays nicely with disjoint union, we know that $$\chi\left(\overline{S}\right) = \chi(S) + \chi(\partial S).$$ If your software can get the first and third terms via $V - E + F$, then you can subtract to get the middle term $\chi(S)$.

Edit: in this case the boundary is a circle, so $\chi(\partial S)=0$. That’d be a nice sanity check for your code!