The Treewidth and Pathwidth of Graph Unions
Abstract.
Given two \(n\)-vertex graphs \(G_1\) and \(G_2\) of bounded treewidth, is there an \(n\)-vertex graph \(G\) of bounded treewidth having subgraphs isomorphic to \(G_1\) and \(G_2\)? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if \(G_1\) is a binary tree and \(G_2\) is a ternary tree. We also provide an extensive study of cases where such “gluing” is possible. In particular, we prove that if \(G_1\) has treewidth \(k\) and \(G_2\) has pathwidth \(\ell\), then there is an \(n\)-vertex graph of treewidth at most \(k+3 \ell+1\) containing both \(G_1\) and \(G_2\) as subgraphs.
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Acknowledgment.
We are very grateful to the anonymous referee for their careful reading of our manuscript, and for providing us with several interesting observations, some of which we included in the conclusion section.
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© 2024 Bogdan Alecu, Vadim V. Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev.
History
Submitted: 23 September 2022
Accepted: 3 October 2023
Published online: 10 January 2024
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Programa Regional: MATH-AMSUD MATH210008
Funding: The work of the third author was supported by FONDECYT/ANID Iniciación en Investigación grant 11201251 and by Programa Regional MATH-AMSUD MATH210008.
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