# Reducing Topological Minor Containment to the Unique Linkage Theorem

In the Topological Minor Containment problem (TMC) problem two undirected graphs, G and H are given and the objective is to check whether G contains H as a topological minor. Grohe, Kawarabayashi, Marx, and Wollan [STOC 2011] resolved the parameterized complexity of this problem by designing an algorithm with running time f(k)n^3. Here, k=|V(H)| and n=|V(G)|. Among several other tools, this algorithm was based on a variant of the linkage theorem of Robertson-Seymour, tailor-made for TMC, by adapting elements of the new proof of the unique linkage theorem of Kawarabayashi and Wollan [STOC 2010]. In this paper we turn the wheel back, and design an alternate algorithm for TMC by giving a direct reduction to a version of unique linkage theorem appearing in Graph Minors XXII of Robertson and Seymour. That is, the new proof requires a black-box invocation of a known result arising in the proof of graph minor theorem. Our result has the following advantages: (1) Graph classes where the bounds in the unique linkage theorem are known, this immediately results in explicit time-bounds for TMC on those graph classes. For example, we get first algorithm with explicit time bounds for TMC on planar graphs and more generally for graphs of genus g. In particular, the algorithm runs in time 2^2^2^ O(k) on planar graphs. (2) Any new development on the unique linkage theorem will immediately led to a faster algorithm for TMC. That is, one does not need to design a similar theorem specifically for TMC.

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