Planar graphs with separation are dp-minimal

TL;DR

This paper proves that expanding a planar graph with predicates for separation by simple cycles results in a dp-minimal structure, linking graph topology with model-theoretic properties.

Contribution

It introduces a novel expansion of planar graphs by cycle separation predicates and establishes dp-minimality of the resulting structure.

Findings

Expansion by cycle separation predicates yields dp-minimal structures.

The result connects graph topology with model-theoretic simplicity.

Provides a new perspective on the logical complexity of planar graphs.

Abstract

We prove that given a planar embedding of a graph in the sphere the expansion of the graph structure by predicates encoding separation of vertices by simple cycles of the graph is dp-minimal.