The book "Basic Topology" by Armstrong has a section on Euler's Polyhedra formula,
he specifically states:
Euler's Theorem let $P$ be a polyhedra which satisfies:
a. Any two vertices of $P$ can be connected by a chain of edges
b. Any loop on $P$ which is made up of straight line segments (not necessarily edges) separates $P$ into $2$ pieces.
Then $v - e + f = 2$.
So I was toying around with polyhedra and I came up with the following example, consider 2 tetrahedra $T_1, T_2$. Connect at a point, call this structure $G$. Then if one applies Euler's formula to this they get $(4+3) - (6+6) + (4+4) = 3$, but every pair of vertices is edge connected, so the only conclusion is that there is a loop on the surface of $G$, which doesn't separate $G$ into 2 parts.
But I am having trouble finding such a loop intuitively, so I decided to try to fill in some details.
We consider a loop $l$ on the surface of $G$ as a connected non self intersecting curve. now we need to make a notion of seperation, the idea I had was to look at finding a surface of minimal surface area $S$ that has the loop $l$ as its boundary. (This hopefully gets close to the notion of "slicing" along a loop)
One can then look at $N_\epsilon(S)$ which is to take the surface of $S$ and at each point form an $\epsilon$ sized ball around it.
Then the claim is that for sufficiently small $\epsilon$
$$ G - N_\epsilon(S) $$
has 2 connected components.
So given this definition, I wanted to try to apply it to the $G$ I constructed but that is starting to get a little messy, so I wanted to call for some backup and ask if theres an obvious way to find the loop.