Every cograph on n positive vertices has a spanning tree of crossing number at most $4 * (ceil(log₂ n) + 1)$ for its open-neighborhood set system.

Indeed, a Welzl order determines the path that joins consecutive vertices. A set crosses exactly the same consecutive pairs in the order as it crosses edges of this path. The logarithmic Welzl-order bound for cographs therefore transfers without loss to spanning trees.

In the terminology of Crespelle and Gambette, the underlying order theorem is their logarithmic upper bound on cograph contiguity: a neighborhood that is the union of few intervals has few membership changes along the order. The path construction records those changes as crossed tree edges.