Algorithms for the edge-width of an embedded graph

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Let G be an unweighted graph of complexity n embedded in a surface of genus g, orientable or not. We describe improved algorithms to compute a shortest non-contractible and a shortest non-separating cycle in G.If k is an integer, we can compute such a non-trivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edge-width or face-width of a graph is bounded from above by a constant. This also implies an output-sensitive algorithm to compute a shortest non-trivial cycle that runs in O(gnk0) time, where k0 is the length of the cycle.We also give an approximation algorithm for the shortest non-trivial cycle. If a parameter 0<ε<1 is given, we compute in O(gn/ε) time a non-trivial cycle whose length is at most 1+ε times the length of the shortest non-trivial cycle.