Rooted prism-minors and disjoint cycles containing a specified edge

TL;DR

This paper characterizes 3-connected graphs with a specific edge that is not part of any pair of disjoint cycles, extending classical results on graph minors and connectivity.

Contribution

It provides a complete characterization of 3-connected graphs with an edge contained in no pair of vertex-disjoint cycles, including applications to related graph classes.

Findings

Characterization of 3-connected graphs with an edge in no disjoint cycles

Extension of classical results on prism-minors and connectivity

Complete description of 3-connected graphs with no prism-minor involving a specified edge

Abstract

Dirac and Lov\'{a}sz independently characterized the $3$-connected graphs with no pair of vertex-disjoint cycles. Equivalently, they characterized all $3$-connected graphs with no prism-minors. In this paper, we completely characterize the $3$-connected graphs with an edge that is contained in the union of no pair of vertex-disjoint cycles. As applications, we answer the analogous questions for edge-disjoint cycles and for $4$-connected graphs and we completely characterize the $3$-connected graphs with no prism-minor using a specified edge.