Well-quasi-orders on embedded planar graphs

TL;DR

This paper extends the theory of graph minors to embedded planar graphs, proving that certain embedded minor relations form well-quasi-orders, which has implications for graph structure analysis and algorithm design.

Contribution

It introduces and proves that embedded minor relations are well-quasi-orders on classes of embedded planar graphs, extending classical results to topologically constrained settings.

Findings

Embedded immersion induces a well-quasi-order on bounded carving-width plane graphs.

Embedded graph minor relation forms a well-quasi-order on plane graphs with bounded branch-width.

The embedded graph minor relation is a well-quasi-order on all plane graphs.

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.