FINDING SHORTEST NON-TRIVIAL CYCLES IN DIRECTED GRAPHS ON SURFACES

Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute shortest non-contractible and shortest surface non-separating cycles in D, generalizing previous results that dealt with undirected graphs. Our first algorithm computes such cycles in O (n(2) log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best general algorithm in the undirected case. It revisits and extends Thomassen's 3-path condition; the technique applies to other families of cycles as well. We also provide more efficient algorithms in special cases, such as graphs with small genus or bounded treewidth, using a divide-and-conquer technique that simplifies the graph while preserving the topological properties of its cycles. Finally, we give an efficient output-sensitive algorithm, whose running time depends on the length of the shortest non-contractible or non-separating cycle.