2026-07-25T09:51:01Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14613 2024-03-06T10:54:26Z ddc:004 open_access

An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes Colin de Verdière, Éric Magnard, Thomas computational topology embedding simplicial complex graph surface fixed-parameter tractability 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)n³-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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Éric Colin de Verdière and Thomas Magnard 2021 Is Part Of LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2021.32 urn:nbn:de:0030-drops-146139 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.32 eng https://creativecommons.org/licenses/by/4.0/legalcode