One-Stroke Dissections of a Convex Polygon

A dissection graph of a convex $n$-gon is the figure formed by the $n$-gon together with $n-3$ of its diagonals, no two of which intersect in the interior of the polygon. Prove that there exists a dissection graph that can be drawn in one closed stroke — starting at a vertex, traversing every segment of the figure exactly once, and returning to the starting vertex — if and only if $3 \mid n$. (The figure shows such a dissection for $n=6$.)