I defined a procedure $\Gamma$ to generate graphs following some rules. The obtained graphs form a class of graphs $\mathcal{G}$. I found out that these graphs are chrodal. In the other hand, I noticed that every chordal graph of order 3 and 4 can be constrcuted by the procedure $\Gamma$.
I implemented an algorithm that generates all $\mathcal{G}$-graphs with $n$ vertices. But I need some algorithm that constructs all the chordal graphs with $n$ vertices in order to make the following conjecture more plausible.
Let $G$ be a graph. Then $G$ belongs to $\mathcal{G}$ if and only if $G$ is chordal.
I want to show that every graph obtained by such an algorithm can also be obtained by the procedure $\Gamma$.
So, my question is: Does any algorithm generating all chordal graphs exist?