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 Dujmovi\'c, 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.

翻译：对于$k \geqslant 0$，我们定义一个简单拓扑图$G$（即在平面上绘制、使得任意两条边至多相交一次（包括端点）的图）为$k$-匹配平面的，如果对于每条边$e \in E(G)$，在$G$中与$e$相交的边所构成的任何匹配的大小至多为$k$。$k$-匹配平面图类是对许多现有超平面图类（包括$k$-平面图）的重要推广。我们证明，每个简单拓扑$k$-匹配平面图都同构于一个有界树宽图与一条路的强积的子图。这一结果在性质上扩展了Dujmovi\'c、Joret、Micek、Morin、Ueckerdt和Wood的平面图积结构定理[J. ACM 2020]以及近期其他超平面图类的积结构定理。利用这一结果，我们推断简单拓扑$k$-匹配平面图类具有若干吸引人的性质，使其成为文献中具有这些性质的最广泛的简单超平面图类。我们关于简单拓扑$k$-匹配平面图的所有结果均可推广到非简单情形，此时与一个公共顶点相关联的成对交叉边的最大数量变得相关。本文引入了若干具有独立意义的工具和结果。我们证明每个简单拓扑$k$-匹配平面图允许使用$\mathcal{O}(k^{3}\log k)$种颜色进行边着色，使得同色边互不交叉。作为我们主要积结构定理证明的关键要素，我们引入了弱浅次图的概念，它包含并推广了图稀疏理论中的关键概念——浅次图。我们还为具有良好圆形画法的图建立了树宽的上界，这在性质上推广了若干现有结果。