A spatial version of Tutte's conflict graph

TL;DR

This paper introduces a spatial version of Tutte's conflict graph, linking flat embeddings of graphs to the balance of associated conflict graphs, and conjectures a characterization of intrinsic linking.

Contribution

It defines a signed conflict graph for maximally planar subgraphs in nonplanar graphs and relates flat embeddings to the balance of these conflict graphs.

Findings

If G has a flat embedding, all conflict graphs are balanced.

Maximal planar subgraphs of G lie on a sphere intersecting G only in that subgraph.

Conjecture: G is intrinsically linked iff all conflict graphs are unbalanced.

Abstract

Tutte showed that a graph $G$ is planar if and only if the conflict graph associated to every cycle of $G$ is bipartite. We define a (not necessarily unique) signed conflict graph associated to a maximally planar subgraph of a nonplanar graph such that if $G$ has a flat embedding, every possible conflict graph associated to every maximally planar subgraph of $G$ is balanced. In doing this, we show that for every graph $G$ with flat embedding, and a planar subgraph $P$ of $G$, $P$ lies on a sphere that intersects $G$ only in $P$. We conjecture that $G$ is intrinsically linked if and only if every maximal planar subgraph of $G$ has every possible conflict graph unbalanced.