Two Analogs of Intrinsically Linked Graphs

TL;DR

This paper characterizes intrinsically S^1-linked graphs as exactly the non-outer-planar graphs and introduces the concept of outer-flat graphs, showing they are precisely the planar graphs.

Contribution

It establishes a new characterization of intrinsically S^1-linked graphs and introduces the concept of outer-flat graphs, linking them to planarity.

Findings

A graph is intrinsically S^1-linked iff it is not outer-planar.

A graph is outer-flat iff it is planar.

Outer-flat graphs can be embedded in a 3-ball with specific boundary conditions.

Abstract

A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embedded in the 3-ball such that all of its vertices map to the boundary of the 3-ball, all edges to the interior, and every cycle bounds a disk in the 3-ball that meets the graph only along its boundary. We show that a graph is outer-flat if and only if it is planar.