# On disjoint cycles

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## Abstract

It is shown, that for each constant *k* ≥ 1, the following problems can be solved in *O*(*n*) time: given a graph *G*, determine whether *G* has *k* vertex disjoint cycles, determine whether *G* has *k* edge disjoint cycles, determine whether *G* has a feedback vertex set of size ≤ *k*. Also, every class \(\mathcal{G}\), that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of *k* copies of *K*_{3}, has an \(\mathcal{O}\)(*n*) membership test algorithm.

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© Springer-Verlag Berlin Heidelberg 1992