An example of a non-hyponormal injective composition operator in an L^2-space generating Stieltjes moment sequences, invented by Jablonski, Jung and JS, was built over a non-locally finite directed tree. The main goal of my talk is to show how to solve the problem of whether there exists such an operator over a locally finite directed graph and, in the affirmative case, to find the simplest possible graph with these properties, where simplicity refers to local valency. It will be shown that the problem can be solved affirmatively for the locally finite directed graph $G_{2,0}$, which consists of two branches and one loop. The only simpler directed graph for which the problem remains unsolved consists of one branch and one loop. The consistency condition, which is the only efficient tool for verifying subnormality of unbounded composition operators, will be disccused in the context of the graph $G_{2,0}$. This will lead to a constructive method of solving the problem. The method itself is partly based on transforming the Krein and the Friedrichs measures coming either from shifted Al-Salam-Carlitz q-polynomials or from a quartic birth and death process.