Binomial edge ideals of cographs

We determine the Castelnuovo--Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). On $n$ vertices the maximum regularity is essentially $2n/3$. Independently of the number of vertices, we also bound the regularity by graph theoretic invariants. Finally, we construct a family of counterexamples to a recent conjecture of Hibi and Matsuda.