Structure of $k$-Matching-Planar Graphs

TL;DR

The paper introduces a product-structure framework for simple topological $k$-matching-planar graphs, proving every such graph embeds in a strong product $H \boxtimes P$ where $H$ has bounded treewidth. Central to the approach are Coloured Planarisations and the novel notion of weak shallow minors, enabling transfer of planar-like structure to beyond-planar graphs and yielding bounds on row and layered treewidth. These structural results imply broad algorithmic and combinatorial consequences, including universal labeling schemes, bounded queue numbers, and controlled nonrepetitive colourings, extending to non-simple graphs as well. The authors further develop bounds for circular drawings and illustrate the versatility of weak shallow minors in graph sparsity, offering tools likely applicable to a wider class of beyond-planar graphs.

Abstract

For $k \geqslant 0$, we define a simple topological graph $G$ (that is, a graph drawn in the plane such that every pair of edges intersect at most once, including endpoints) to be $k$-matching-planar if for every edge $e \in E(G)$, every matching amongst the edges of $G$ that cross $e$ has size at most $k$. The class of $k$-matching-planar graphs is a significant generalisation of many other existing beyond planar graph classes, including $k$-planar graphs. We prove that every simple topological $k$-matching-planar graph is isomorphic to a subgraph of the strong product of a graph with bounded treewidth and a path. This result qualitatively extends the planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] and recent product structure theorems for other beyond planar graph classes. Using this result, we deduce that the class of simple topological $k$-matching-planar graphs has several attractive properties, making it the broadest class of simple beyond planar graphs in the literature that has these properties. All of our results about simple topological $k$-matching-planar graphs generalise to the non-simple setting, where the maximum number of pairwise crossing edges incident to a common vertex becomes relevant. The paper introduces several tools and results of independent interest. We show that every simple topological $k$-matching-planar graph admits an edge-colouring with $\mathcal{O}(k^{3}\log k)$ colours such that monochromatic edges do not cross. As a key ingredient of the proof of our main product structure theorem, we introduce the concept of weak shallow minors, which subsume and generalise shallow minors, a key concept in graph sparsity theory. We also establish upper bounds on the treewidth of graphs with well-behaved circular drawings that qualitatively generalise several existing results.

Figures (7)

• Figure 1: (a) $K_{3, n}$ is $1$-matching-planar. (b) A topological $1$-matching-planar graph, where every edge crosses $n$ edges.
• Figure 2: Forbidden crossing configurations. Configuration I: $e$ is crossed by two edges that are not incident to a common vertex. Configuration II: $e$ is crossed by two edges that cross $e$ from different sides when directed away from a common endpoint. Configuration III: both endpoints of $e$ are in the bounded region determined by $e$ and two edges that cross $e$ and are incident to a common vertex. Configuration IV: $e$ is crossed by the edges of a triangle.
• Figure 3: An example of a planarisation and a coloured planarisation. (a) A topological graph $G$ isomorphic to $K_{3, 5}$ with a transparent ordered $2$-edge-colouring $\phi$, where colours are: red $= 1$, blue $= 2$. (b) The planarisation $G'$ of $G$ where every dummy vertex has level $1$ and every vertex of $G$ has level $0$. (c) The coloured planarisation $G^{\phi}$ of $G$ obtained by contracting red edges of $G'$ not incident to $V(G)$. Every vertex of $V(G^{\phi}) \setminus V(G)$ has level $1$ and every vertex of $G$ has level $0$.
• Figure 4: An example of fragments and sections. (a) A topological graph $G$ with a transparent ordered $5$-edge-colouring $\phi$, where colours are: green $= 1$, blue $= 2$, black $= 3$ (only the edge $ab$ is black), red $= 4$, and brown $= 5$. The edge $ab$ is split by the crossing points (marked as squares) of $ab$ and the edges of smaller colours into five fragments (highlighted in purple). (b) The planarisation $G'$ of $G$. Each vertex is labelled by its level. The edges of sections and $1$-vertex sections of $G'$ are highlighted in purple. There are three sections of $L_{ab}$, one of which consists of a single dummy vertex labelled $d$.
• Figure 5: The coloured planarisation $G^{\phi}$ of the graph $G$ with the transparent ordered $5$-edge-colouring $\phi$ from \ref{['fragmentssectionsa']}. Each vertex is labelled by its level. The edges between consecutive vertices of the walk $W_{ab}$ in $G^{\phi}$ are highlighted in purple.

Theorems & Definitions (106)

• Theorem 1.1: Planar Graph Product Structure Theorem DJMMUW20
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: DMW17
• Lemma 1.9
• Theorem 1.10