Binomial Edge Ideals of K\"onig Type

We first characterise graphs with binomial edge ideals of König type as those for which the path covering number is equal to a minor variant of the scattering number.
This enables us to apply known graph-theoretic results to immediately deduce that several classes of graphs have binomial edge ideals of König type. In particular, we show this for cocomparability graphs, or weakly closed graphs in the language of Matsuda.
Along with work of LaClair and McCullough, this allows us to prove that an unmixed binomial edge ideal is of König type if and only if G is weakly closed.
We then conjecture that AT-free graphs have binomial edge ideals of König type.