Topological graph theory
A new approach to the maximum-flow problem
Journal of the ACM (JACM)
Analysis of preflow push algorithms for maximum network flow
SIAM Journal on Computing
Maximum (s,t)-flows in planar networks in O(|V| log |V|) time
Journal of Computer and System Sciences
On determining the genus of a graph in O(v O(g)) steps(Preliminary Report)
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Isomorphism testing for graphs of bounded genus
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
An O (n log n) algorithm for maximum st-flow in a directed planar graph
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Homology flows, cohomology cuts
Proceedings of the forty-first annual ACM symposium on Theory of computing
Minimum cuts and shortest homologous cycles
Proceedings of the twenty-fifth annual symposium on Computational geometry
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There is a large body of results on planar graph algorithms that are more efficient than the best known algorithm for general graphs [13]. Maximum flow [1] is but one example. More drastically, the maximum cut problem is polynomially solvable for planar instances but NP-complete in general [12, 8, 5]. However, little is known about nearly planar graphs. This is unsatisfactory since the nearly planar case is particularly important in practice. Think for example of road networks with bridges and tunnels. We present a preflow push algorithm that solves the maximum s-t-flow problem in a network with n vertices and m edges and embedded with k crossings in time O(k3n log n) worst case. To our knowledge there is only one previous result that relates asymptotic running time to a topological parameter of the graph such that the running time is polynomial in this parameter. Compared with the currently fastest maximum flow algorithms this reduces the worst case running time by a factor of m/k3 ignoring logarithmic factors. Therefore, it is particularly favorable for very sparse or nearly planar graphs.