In 2020, Bonnet et al. invented the graph invariant "twin-width" relying on contraction sequences for graphs and showed that on graphs of bounded twin width many classical graph algorithms become tractable.

In 2025, Ahn et al. showed that twin-width 1 graphs are permutation graphs and gave an algorithm to compute the contraction sequence efficiently, which follows closely the permutation diagram.

We will reverse-engineer this algorithm to generate permutation diagrams covering all twin-width 1 graphs (many of multiply) using the symbolic method. Eventually, this gives us bounds for the number of twin-width 1 graphs of order $c^n n!$ with explicit $c$ for the upper bound. Generating twin-width 1 graphs from particular graph classes whose counting sequence is known, we get a lower
bound of the same kind.

Joint work with Zbigniew Gołębiewski, Małgorzata Sulkowska and Michael Wallner.