Yes. We induct on the number of sides $n$ of the polygon. $n=3$ is trivial. We can assume it is regular. Consider the smallest diagonal, and note that it must divide the polygon into a triangle and a $(n-1)$-gon. So there is a vertex with number 1 assigned to it. So we can draw the smallest diagonal that just cuts out that vertex. Decrease the number of the vertices connected by that diagonal by 1. Now we are left with a $(n-1)$-gon, whose diagonals can be uniquely reconstructed by our induction hypothesis, and we are done.