Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomasse's recent proof of Gallai's conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent. We then derive short proofs of Gallai's conjecture and a theorem "polar to" the main result of Bessy and Thomasse, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.