The treewidth and pathwidth of graph unions



Alecu, Bogdan, Lozin, Vadim, Quiroz, Daniel A, Rabinovich, Roman, Razgon, Igor and Zamaraev, Viktor ORCID: 0000-0001-5755-4141
(2022) The treewidth and pathwidth of graph unions. [Report]

[img] Text
2202.07752v1.pdf - Submitted Version

Download (235kB) | Preview

Abstract

For two graphs $G_1$ and $G_2$ on the same vertex set $[n]:=\{1,2, \ldots, n\}$, and a permutation $\varphi$ of $[n]$, the union of $G_1$ and $G_2$ along $\varphi$ is the graph which is the union of $G_2$ and the graph obtained from $G_1$ by renaming its vertices according to $\varphi$. We examine the behaviour of the treewidth and pathwidth of graphs under this "gluing" operation. We show that under certain conditions on $G_1$ and $G_2$, we may bound those parameters for such unions in terms of their values for the original graphs, regardless of what permutation $\varphi$ we choose. In some cases, however, this is only achievable if $\varphi$ is chosen carefully, while yet in others, it is always impossible to achieve boundedness. More specifically, among other results, we prove that if $G_1$ has treewidth $k$ and $G_2$ has pathwidth $\ell$, then they can be united into a graph of treewidth at most $k + 3 \ell + 1$. On the other hand, we show that for any natural number $c$ there exists a pair of trees $G_1$ and $G_2$ whose every union has treewidth more than $c$.

Item Type: Report
Uncontrolled Keywords: math.CO, math.CO, cs.DM
Divisions: Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science
Depositing User: Symplectic Admin
Date Deposited: 23 Feb 2022 10:39
Last Modified: 04 May 2022 02:10
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3149491