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Communication Dans Un Congrès Année : 2021

An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes

Résumé

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f (c)n O(c) , where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f (c)n 3-time algorithm. We also provide a randomized algorithm with expected running time 2 c O(1) n O(1). Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.
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Dates et versions

hal-03372522 , version 1 (10-10-2021)

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Éric Colin de Verdière, Thomas Magnard. An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes. European Symposium on Algorithms, Sep 2021, Lisbon, Portugal. ⟨10.4230/LIPIcs.ESA.2021.32⟩. ⟨hal-03372522⟩
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