Bounding Twin-Width for Bounded-Treewidth Graphs, Planar Graphs, and Bipartite Graphs

Abstract

Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of “nicely structured” graph classes. In this work, we focus on obtaining good bounds on twin-width $\textbf{tww}(G)$ for graphs G from a number of classic graph classes. We prove the following:

• $\textbf{tww}(G) \le 3\cdot 2^{\textbf{tw}(G)-1}$, where $\textbf{tw}(G)$ is the treewidth of G,

• $\textbf{tww}(G) \le \max (4\textbf{bw}(G),\frac{9}{2}\textbf{bw}(G)-3)$ for a planar graph G with $\textbf{bw}(G) \ge 2$, where $\textbf{bw}(G)$ is the branchwidth of G,

• $\textbf{tww}(G) \le 183$ for a planar graph G,

• the twin-width of a universal bipartite graph $(X,2^X,E)$ with $|X|=n$ is $n - \log _2(n) + \mathcal {O}(1)$.

An important idea behind the bounds for planar graphs is to use an embedding of the graph and sphere-cut decompositions to obtain good bounds on neighbourhood complexity.

This research is part of projects that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreement 714704. Initial part of the reseach was done when Hugo Jacob was on an internship at University of Warsaw in Spring and Summer 2021. The authors acknowledge support from the ERC starting grant “CRACKNP” (Grant Agreement 853234) for attending the conference.

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