Well-quasi-orders on embedded planar graphs

TL;DR

This work extends the graph minor paradigm to embedded plane graphs by introducing and harnessing embedded versions of minor and immersion relations. It develops bond-linked and disc carving-decompositions to support Nash-Williams-type wqo arguments in the bounded carving-width setting, and then transfers the results to plane graphs via medial-graph correspondences and classical grid theorems to address unbounded width. The main achievements are: (i) a well-quasi-order for embedded immersions on plane graphs with bounded carving-width, (ii) a transfer to embedded minors on plane graphs with bounded branch-width through medial graphs, and (iii) a completion of the embedded minor wqo for all plane graphs using unbounded-width grid arguments. These results provide structural and algorithmic consequences for intrinsically embedded objects and contribute to the foundational understanding of embedding-aware graph minor theory.

Abstract

The central theorem of topological graph theory states that the graph minor relation is a well-quasi-order on graphs. It has far-reaching consequences, in particular in the study of graph structures and the design of (parameterized) algorithms. In this article, we study two embedded versions of classical minor relations from structural graph theory and prove that they are also well-quasi-orders on general or restricted classes of embedded planar graphs. These embedded minor relations appear naturally for intrinsically embedded objects, such as knot diagrams and surfaces in $\mathbb{R}^3$. Handling the extra topological constraints of the embeddings requires careful analysis and extensions of classical methods for the more constrained embedded minor relations. We prove that the embedded version of immersion induces a well-quasi-order on bounded carving-width plane graphs by exhibiting particularly well-structured tree-decompositions and leveraging a classical argument on well-quasi-orders on forests. We deduce that the embedded graph minor relation defines a well-quasi-order on plane graphs via their directed medial graphs, when their branch-width is bounded. We conclude that the embedded graph minor relation is a well-quasi-order on all plane graphs, using classical grids theorems in the unbounded branch-width case.

Figures (33)

• Figure 1: Two plane graphs $G$ and $G'$ that are isomorphic but not equivalent as plane graphs. The graph $H$ is a minor of both $G$ and $G'$, an embedded minor of $G$ obtained by contracting the embedding of $e$. But it is not an embedded minor of $G'$: no embedded edge contraction or deletion in $G'$ results in a plane graph with $2$ faces, one of which contains edges in its inside.
• Figure 2: Left: edge deletion of $e$. Right: vertex deletion of $v$. Below: edge contraction of $e$.
• Figure 3: Left: illustration for an embedded lift of two edges $e,e'$ incident to $v$ yielding a merged edge $e"$. Right: Other edges determine which arc is valid for a planar embedding.
• Figure 4: Two embedded graphs, the order around each vertex is induced by the embedding. $G_1$ is an ordered minor of $G_2$ but not an embedded minor of $G_2$.
• Figure 5: Diverse paths of a family of paths (red, blue, and green paths) around a vertex. Left: tangent paths around a vertex and the associated embedded lift. Right: transverse paths around a vertex and the associated lift which is not an embedded lift.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Corollary 1.5
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Theorem 4.1