Can a directed graph have Ω(n^2) cross edges with respect to its DFS tree?

Question

My assumption is that we could have Ω(n^2) cross edges. Does someone have any idea how to prove it?

Answer 1

Yes. Take the acyclic tournament on n vertices (n vertices, n choose 2 arcs, no cycles). Root the traversal at the vertex with out-degree n-1 and visit the nodes with out-degrees 0, 1, 2, ..., n-2 in order. Every non-tree arc is a cross arc.