# On linear time minor tests and depth first search

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## Abstract

Recent results on graph minors make it desirable to have efficient algorithms, that for a fixed set of graphs {*H*_{1}, ..., *H*_{ c }}, test whether a given graph *G* contains at least one graph *H*_{ i } as a minor. In this paper we show the following result: if at least one graph *H*_{ i } is a minor of a 2 × *k* grid graph, and at least one graph *H*_{ i } is a minor of a circus graph, then one can test in \(\mathcal{O}\)(*n*) time whether a given graph *G* contains at least one graph *H*∈{*H*_{1}, ..., *H*_{ c }} as a minor. This result generalizes a result of Fellows and Langston. The algorithm is based on depth first search and on dynamic programming on graphs with bounded treewidth. As a corollary, it follows that the MAXIMUM LEAF SPANNING TREE problem can be solved in linear time for fixed *k*. We also discuss that with small modifications, an algorithm of Fellows and Langston can be modified to an algorithm that finds in \(\mathcal{O}\)(*k*!2^{ k }*n*) time a cycle (or path) in a given graph with length≥*k* if it exists.

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