Agile Sets in Graphs

TL;DR

This work addresses when a graph containing a large agile set must contain a minor of the form $K_{2,k}$ or an associated obstruction. It develops a structural framework combining nested 2-separations, tree-decompositions, and Ding's theorem on $K_{2,k}$-minor-free graphs, together with the augmentation-by-fans-and-strips machinery to derive a dichotomy: large agile sets force either a $K_{2,k}$ minor or a regular strip minor, quantified by a bounding function $f(k)$. The results establish that the phenomenon holds for $k\le 4$ but requires the regular-strip obstruction for larger $k$, and are extended to $m$-agile and dexterous variants via generalized grid theorems and wheel-like constructions. Overall, the paper connects agile-set structure to classical minor theory and grid-like obstructions, yielding a broad qualitative characterization with potential implications for graph density and minor-exclusion 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.

Figures (1)

• Figure 1: The type of graph of which $K$ is a minor.

Theorems & Definitions (46)

• Theorem 1.3
• Lemma 2.1: cf. confing*Theorem 4.8
• Lemma 2.2
• proof
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 4.1