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@article{Diestel1995GraphM1, title={Graph Minors 1: A Short Proof of the Path-width Theorem}, author={Reinhard Diestel}, journal={Combinatorics, Probability & Computing}, year={1995}, volume={4}, pages={27-30} }

- Published 1995 in Combinatorics, Probability & Computing
DOI:10.1017/S0963548300001450

The authors remark that this result is best possible in two ways. First, the value of \F\ — 1 is sharp, because the complete graph Kn-\ has path-width n — 2 but has no (forest) minor on n vertices. Second, if F is not a forest, then the exclusion of F as a minor does not bound the path-width of a graph: as noted without proof in [1], trees can have arbitrarily large path-width (but will never contain F as a minor if F contains a cycle). The proof of Theorem 1 in [2], already much shorter than… CONTINUE READING

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