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Given an undirected, edge-weighted graph G together with pairs of vertices,called pairs of terminals, the minimum multicut problem asks for aminimum-weight set of edges such that, after deleting these edges, the twoterminals of each pair belong to different connected components of the graph.Relying on topological techniques, we provide a polynomial-time algorithm forthis problem in the case where G is embedded on a fixed surface of genus g(e.g., when G is planar) and has a fixed number t of terminals. The runningtime is a polynomial of degree O(sqrt{g^2+gt}) in the input size.  In the planar case, our result corrects an error in an extended abstract byBentz [Int. Workshop on Parameterized and Exact Computation, 109-119, 2012].The minimum multicut problem is also a generalization of the multiway cutproblem, a.k.a. multiterminal cut problem; even for this special case, nodedicated algorithm was known for graphs embedded on surfaces.

Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals