Agile Sets in Graphs

TL;DR

This paper characterizes the existence of large agile vertex sets in graphs, linking them to specific minors like $K_{2,k}$ and large strip minors, advancing understanding of graph connectivity properties.

Contribution

It provides a new characterization for large agile sets in graphs based on minors such as $K_{2,k}$ and strip minors, connecting structural graph theory with connectivity concepts.

Findings

Large agile sets are characterized by the presence of $K_{2,k}$ minors.

Large strip minors also indicate the existence of large agile sets.

The results deepen understanding of graph connectivity and minor theory.

Abstract

A set of vertices in a graph is agile if, however we partition the set into two parts, we can always find two vertex-disjoint connected subgraphs where one covers the first and the other the second part. We present a characterization for the existence of large agile sets in terms of $K_{2,k}$ and large strip minors.