Crossing number additivity over edge cuts

Consider a graph G with a minimal edge cut F and let G 1 , G 2 be the two (augmented) components of G − F . A long-open question asks under which conditions the crossing number of G is (greater than or) equal to the sum of the crossing numbers of G 1 and G 2 —which would allow us to consider those graphs separately. It is known that crossing number is additive for | F | ∈ { 0 , 1 , 2 } and that there exist graphs violating this property with | F | ≥ 4 . In this paper, we show that crossing number is additive for | F | = 3 , thus closing the final gap in the question. chniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing-critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.