Let  G  be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest non-separating cycle in G in O(g 2  n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest non-contractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.
['Jeff Erickson']
Shortest non-trivial cycles in directed surface graphs