[Paper Review] Improved product structure for graphs on surfaces

Marc Distel, Robert Hickingbotham+2 more|arXiv (Cornell University)|Dec 18, 2021

Advanced Graph Theory Research4 citations

TL;DR

This paper improves the product structure theorem for graphs embedded on surfaces by reducing the treewidth bound of the planar graph component from 4 to 3, while maintaining containment in a strong product with a path and a complete graph. The key result shows that every graph of Euler genus $ g $ is contained in $ H \boxtimes P \boxtimes K_{\max\{2g,3\}} $, where $ H $ is a planar graph of treewidth 3, significantly tightening the structural characterization of graphs on surfaces.

ABSTRACT

Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve this result by replacing "4" by "3" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} floor,d+3\lfloor\frac{d}{2} floor-3\}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.

Motivation & Objective

• To refine the structural characterization of graphs embeddable on surfaces using product decompositions.
• To reduce the treewidth bound of the planar component in the product structure theorem from 4 to 3.
• To generalize the result to framed graphs and map graphs on surfaces with bounded face degree.
• To establish optimal bounds for treewidth in the context of product structure theorems for surface-embedded graphs.
• To provide a unified framework for understanding the global structure of graphs on surfaces via strong graph products.

Proposed method

• Introduce the concept of $(g,d)$-framed multigraphs, where edges are added across faces of bounded cycle length in a surface-embedded multigraph.
• Prove that every $(g,d)$-framed multigraph is contained in $ H \boxtimes P \boxtimes K_\ell $, where $ H $ is planar with treewidth 3 and $ \ell = \max\{2g\lfloor d/2 \rfloor, d + 3\lfloor d/2 \rfloor - 3\} $.
• Use edge-maximal embeddings and face cycle analysis to construct a base multigraph $ G_0 $ embedded in a surface of Euler genus $ g $, such that the original graph is a subgraph of $ G_0^{(d)} $.
• Apply the product structure theorem to the resulting framed graph, leveraging the bounded face size and planarity of the base graph to control the treewidth of $ H $.
• Establish that the treewidth bound of 3 is optimal by referencing a known lower bound from Dujmović et al. (2020b).
• Use duality and face cycle reconstruction to embed map graphs and 1-planar graphs as subgraphs of framed graphs, enabling application of the main theorem.

Experimental results

Research questions

• RQ1 Can the treewidth bound in the product structure theorem for graphs on surfaces be reduced from 4 to 3?
• RQ2 Is it possible to replace the apex graph $ H $ in the original theorem with a planar graph of treewidth 3?
• RQ3 What is the optimal value of $ \ell $ such that every $(g,d)$-framed multigraph embeds in $ H \boxtimes P \boxtimes K_\ell $ with $ H $ planar and treewidth 3?
• RQ4 How do map graphs and 1-planar graphs on surfaces relate to framed graphs, and can they be embedded in such product structures?
• RQ5 Is the treewidth bound of 3 for $ H $ optimal in the product structure theorem for surface-embedded graphs?

Key findings

• The paper establishes that every graph of Euler genus $ g $ is contained in $ H \boxtimes P \boxtimes K_{\max\{2g,3\}} $, where $ H $ is a planar graph of treewidth 3, improving the prior bound of treewidth 4.
• The result is generalized to $(g,d)$-framed multigraphs, with $ \ell = \max\{2g\lfloor d/2 \rfloor, d + 3\lfloor d/2 \rfloor - 3\} $, showing that such graphs embed in the same product structure with the same treewidth bound on $ H $.
• Every $(g,1)$-planar graph is contained in $ H \boxtimes P \boxtimes K_{\max\{4g,7\}} $ for some planar $ H $ of treewidth 3, providing a tight structural decomposition.
• The treewidth bound of 3 for $ H $ is optimal, as shown by a known lower bound: for every $ \ell \geq 0 $, there exists a planar graph that requires $ H $ of treewidth at least 3 to embed in $ H \boxtimes P \boxtimes K_\ell $.
• Map graphs on surfaces of Euler genus $ g $ with maximum face degree $ d $ are shown to be subgraphs of $ (g,d) $-framed multigraphs, which are then embedded in the product structure, extending the applicability of the result.
• The construction uses edge-maximal embeddings and face cycle analysis to ensure that the base multigraph $ G_0 $ has all faces bounded by cycles, enabling the use of framed graph theory to embed the original graph in the product.