I'ven been solving problems from my Topology course, and don't understand something I saw while reading my solved examples. Here's a problem that will let me show my point:
Given $X$ a compact surface for which $$a\phantom{i}b\phantom{i}c\phantom{i}a^{-1}\phantom{i}b^{-1}\phantom{i}c^{-1}$$ is a canonical representation. Describe the structure of this surface, calculate it's Euler characteristic and it's orientability type. Classify the surface and give a standar canonical representation of it.
In the solution included with the exercise, it's assumed there are just two vertices, while when I tried to do it myself before spoiling me by seeing the solution the intuition clearly tells there are $3$ vertices, the one between edges $a,b$ which I called $1$, the one between edges $b,c$ which I called $2$ and the one between vertices $a,c$ which I called $3$.
Instead of that, the given solution just assigns vertices $1$ and $2$ alternating them in the drawing. How is it known that there are just $2$ vertices? Is it always just $2$ vertices what I need when given this kind of representation? That number clearly matters since Euler's characteristic depends partially on the number of vertex.
After having that detail clarified, the rest of the questions are solvable by myself, I just wanna know how can I tell how many vertices should I consider when solving an exercise of this kind, where a representation like that is given. Any help or hint will be appreciated, thanks in advance.
