Erd\H{o}s-P\'osa property for labelled minors: 2-connected minors

TL;DR

This paper extends the Erdős-Pósa property to labelled minors, characterizing 2-connected labelled graphs for which a packing-covering duality holds, building on Robertson and Seymour's work on planar graphs.

Contribution

It provides a characterization of 2-connected labelled graphs that satisfy the Erdős-Pósa property for minors, generalizing previous results to labelled graph settings.

Findings

Characterization of 2-connected labelled graphs with Erdős-Pósa property

Extension of duality results to labelled minors

Connection to Robertson and Seymour's planar graph results

Abstract

In the 1960s, Erd\H{o}s and P\'osa proved that there is a packing-covering duality for cycles in graphs. As part of the graph minor project, Robertson and Seymour greatly extended this: there is such a duality for $H$-expansions in graphs if and only if $H$ is a planar graph (this includes the previous result for $H=K_3$). We consider vertex labelled graphs and minors and provide such a characterisation for $2$-connected labelled graphs $H$.