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• W4313227645 abstract "We provide a “vortex-free” refinement of the seminal structure theorem for $K_{t} -$minor free graphs by Robertson and Seymour as follows: we identify a (parameterized) graph H <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>t</inf> and we prove that if we replace K <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>t</inf> by H <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>t</inf> , then the resulting decomposition becomes “vortex-free”. Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the single-crossing minor-free graphs. This result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some H <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>t</inf> , the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an $H_{t} -$minor-free graph G, computes the generating function of all perfect matchings of G in polynomial time. This algorithm yields, on $H_{t} -$minor-free graphs, polynomial algorithms for computational problems such as the dimer problem, the exact matching problem, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graph classes where the number of perfect matchings is polynomially computable: They are precisely those graph classes that do not contain every H <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>t</inf> as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes." @default.
• W4313227645 created "2023-01-06" @default.
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• W4313227645 date "2022-10-01" @default.
• W4313227645 modified "2023-09-24" @default.
• W4313227645 title "Killing a vortex" @default.
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• W4313227645 doi "https://doi.org/10.1109/focs54457.2022.00104" @default.
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