Abstract
We show that the optimization versions of the Pathwidth and Treewidth problems have a property called finite integer index when the inputs are restricted to graphs of bounded pathwidth and bounded treewidth, respectively. They do not have this property in general graph classes. This has interesting consequences for kernelization of both these (optimization) problems on certain sparse graph classes. In the process we uncover an interesting property of path and tree decompositions, which might be of independent interest.
Research funded by DFG-Project RO 927/12-1 “Theoretical and Practical Aspects of Kernelization”, by the Czech Science Foundation, project GA14-03501S, and by Employment of Newly Graduated Doctors of Science for Scientific Excellence (CZ.1.07/2.3.00/30.0009).
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A modulator to treedepth \(d\) of graph \(G\) is a set \(X\subseteq V(G)\) s.t. the treedepth of \(G-X\) is at most \(d-1\). Modulators to bounded treewidth and pathwidth are defined similarly.
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Acknowledgement
We thank Hans L. Bodlaender for valuable discussions about the properties of characteristics of path and tree decompositions.
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Gajarský, J. et al. (2014). Finite Integer Index of Pathwidth and Treewidth. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_22
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DOI: https://doi.org/10.1007/978-3-319-13524-3_22
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