Campbell, Rutger;
Clinch, Katie;
Distel, Marc;
Gollin, J Pascal;
Hendrey, Kevin;
Hickingbotham, Robert;
Huynh, Tony;
... Wood, David R; + view all
(2023)
Product structure of graph classes with bounded treewidth.
Combinatorics, Probability and Computing
pp. 1-26.
10.1017/s0963548323000457.
(In press).
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Abstract
We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class G to be the minimum non-negative integer c such that, for some function f, for every graph G∈G there is a graph H with tw(H)⩽c such that G is isomorphic to a subgraph of H⊠Kf(tw(G)). We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of Ks,t-minor-free graphs has underlying treewidth s (for t⩾max{s,3}); and the class of Kt-minor-free graphs has underlying treewidth t−2. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no H subgraph has bounded underlying treewidth if and only if every component of H is a subdivided star, and that the class of graphs with no induced H subgraph has bounded underlying treewidth if and only if every component of H is a star.
| Type: | Article |
|---|---|
| Title: | Product structure of graph classes with bounded treewidth |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1017/s0963548323000457 |
| Publisher version: | https://doi.org/10.1017/S0963548323000457 |
| Language: | English |
| Additional information: | © The Author(s) 2023. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/). |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10183651 |
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