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Product structure of graph classes with bounded treewidth

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Title
Product structure of graph classes with bounded treewidth
Author(s)
Rutger Campbell; Clinch, Katie; Distel, Marc; J. Pascal Gollin; Kevin Hendrey; Hickingbotham, Robert; Huynh, Tony; Illingworth, Freddie; Tamitegama, Youri; Tan, Jane; Wood, David R.
Publication Date
2024-05
Journal
COMBINATORICS PROBABILITY & COMPUTING, v.33, no.3, pp.351 - 376
Publisher
CAMBRIDGE UNIV PRESS
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 $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$. 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 $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$); and the class of $K_t$-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.
URI
https://pr.ibs.re.kr/handle/8788114/15036
DOI
10.1017/S0963548323000457
ISSN
0963-5483
Appears in Collections:
Pioneer Research Center for Mathematical and Computational Sciences(수리 및 계산과학 연구단) > Discrete Mathematics Group(이산 수학 그룹) > 1. Journal Papers (저널논문)
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