Intrinsically spherical 3-linked graphs

TL;DR

This paper identifies specific families of planar graphs that are minimally intrinsically spherical 3-linked, meaning they always contain a non-split 3-link in any spherical embedding, and conjectures a complete classification of such graphs.

Contribution

It introduces new families of minor-minimal intrinsically spherical 3-linked graphs and proposes a conjecture for their complete classification.

Findings

Identified several families of minor-minimal intrinsically spherical 3-linked graphs.

Proposed a conjecture for the complete set of such graphs involving specific unions of complete and bipartite graphs.

Established the concept of intrinsic spherical 3-linking in planar graphs.

Abstract

We exhibit several families of planar graphs that are minor-minimal intrinsically spherical $3$-linked. A graph is intrinsically spherical 3-linked if it is planar graph that has, in every spherical embedding, a non-split 3-link consisting of two disjoint cycles ($S^1$s) and two disjoint vertices ($S^0$), or a cycle and two pairs of disjoint vertices. We conjecture that $K_4 \dot{\bigcup} K_4$, $K_{3,2} \dot{\bigcup} K_{3,2}$, and $K_4 \dot{\bigcup} K_{3,2}$ form the complete set of minor-minimal intrinsically type I spherical 3-linked graphs (that is, in every spherical embedding, have a nonsplit link of two cycles and one $S^0$).