I don't have an answer to your main question but I can speak to some of your other points.

Firstly, Alperin diagrams are no so much graphs as lattices. They are in fact an ordering on the composition factors of $M$ such that $N_1 \ge N_2$ iff there is a submodule with head $N_1$ and socle $N_2$. As such, triangles (as @alex-dugas notes) are not possible - $N_1 \ge N_2 \ge N_3$ and $N_3 \ge N_1$ are incompatible.

Secondly, a very important restriction on $M$ that must hold before we can construct Alperin diagrams is that $M$ must be multiplicity free. That is, each factor must appear in a composition series exactly once.

To see why this is important, consider two identical modules with composition factors $[N_1, N_2]$ (so the head is $N_1$). Then the direct sum of these modules, quotiented out by a suitable diagonal image of $N_2$ will have head $N_1\oplus N_1$ and socle $N_2$. What is its Alperin diagram? At first we may say

N_1   N_1
   \ /
   N_2

but in fact a suitable change of basis gives that $N_1$ falls out as a direct summand! Thus a "better" diagram is the forest

         N_1
N_1  +    |
         N_2

Which should we use? This is a simple case, but you can imagine that in a module with a dozen factors some of which are repeated multiple times, this becomes a serious question.

I don't have an answer to your main question but I can speak to some of your other points.

Firstly, Alperin diagrams are not so much graphs as lattices. They are in fact an ordering on the composition factors of $M$ such that $N_1 \ge N_2$ iff there is a submodule with head $N_1$ and socle $N_2$. As such, triangles (as @AlexDugas notes) are not possible — $N_1 \ge N_2 \ge N_3$ and $N_3 \ge N_1$ are incompatible.

Secondly, a very important restriction on $M$ that must hold before we can construct Alperin diagrams is that $M$ must be multiplicity free. That is, each factor must appear in a composition series exactly once.

To see why this is important, consider two identical modules with composition factors $[N_1, N_2]$ (so the head is $N_1$). Then the direct sum of these modules, quotiented out by a suitable diagonal image of $N_2$ will have head $N_1\oplus N_1$ and socle $N_2$. What is its Alperin diagram? At first we may say

N_1   N_1
   \ /
   N_2

but in fact a suitable change of basis gives that $N_1$ falls out as a direct summand! Thus a "better" diagram is the forest

         N_1
N_1  +    |
         N_2

Which should we use? This is a simple case, but you can imagine that in a module with a dozen factors some of which are repeated multiple times, this becomes a serious question.