# Decompositions of critical trees with cutwidth *k*

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## Abstract

The cutwidth of a graph *G* is the minimum integer *k* such that the vertices of *G* are arranged in a linear layout \([v_1,\ldots ,v_n]\) in such a way that, for every \(j= 1,\ldots ,n-1\), there are at most *k* edges with one endpoint in \(\{v_1,\ldots ,v_j\}\) and the other in \(\{v_{j+1},\ldots ,v_n\}\). The cutwidth problem for *G* is to determine the cutwidth *k* of *G*. A graph *G* with cutwidth *k* is *k*-cutwidth critical if every proper subgraph of *G* has cutwidth less than *k* and *G* is homeomorphically minimal. In this paper, we obtain that any *k*-cutwidth critical tree \(\mathcal {T}\) can be decomposed into three \((k-1)\)-cutwidth critical subtrees for \(k\ge 2\); And an \(O(|V(\mathcal {T})|^{2}\mathrm{log} |V(\mathcal {T})|)\) algorithm of computing the three subtrees is given.

## Keywords

Graph labeling Cutwidth Critical tree Decomposition## Mathematics Subject Classification

05C75 05C78 90C27## Notes

### Acknowledgements

The author would like to thank the referees for their helpful suggestions on improving the representation of this paper.

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