Computing Tutte Paths

Abstract: Tutte paths are one of the most successful tools for attacking Hamiltonicity problems in planar graphs. Unfortunately, results based on them are non-constructive, as their proofs inherently use an induction on overlapping subgraphs and these overlaps hinder to bound the running time to a polynomial. For special cases however, computational results of Tutte paths are known: For 4-connected planar graphs, Tutte paths are in fact Hamiltonian paths and Chiba and Nishizeki showed how to compute such paths in linear time. For 3-connected planar graphs, Tutte paths have a more complicated structure, and it has only recently been shown that they can be computed in polynomial time. However, Tutte paths are defined for general 2-connected planar graphs and this is what most applications need. Unfortunately, no computational results are known. We give the first efficient algorithm that computes a Tutte path (for the general case of 2-connected planar graphs). One of the strongest existence results about such Tutte paths is due to Sanders, which allows to prescribe the end vertices and an intermediate edge of the desired path. Encompassing and strengthening all previous computational results on Tutte paths, we show how to compute this special Tutte path efficiently. Our method refines both, the results of Thomassen and Sanders, and avoids overlapping subgraphs by using a novel iterative decomposition along 2-separators. Finally, we show that our algorithm runs in quadratic time.