Rankwidth of Graphs with Balanced Separations: Expansion for Dense Graphs

TL;DR

This paper establishes a new relationship between rankwidth and well-linkedness in graphs, showing that dense graphs with high rankwidth contain well-connected subgraphs, and introduces rank-expansion as a measure of expansion.

Contribution

It proves that graphs with high rankwidth contain induced subgraphs with large balanced cutrank, linking rankwidth to well-linkedness and supporting rank-expansion as an expansion measure.

Findings

Graphs with rankwidth ≥ 72r contain subgraphs with cutrank ≥ r

High rankwidth implies existence of well-linked subgraphs

Supports rank-expansion as a measure of dense graph expansion

Abstract

We prove that every graph of rankwidth at least $72r$ contains an induced subgraph whose minimum balanced cutrank is at least $r$, which implies a vertex subset where every balanced separation has $\mathbb{F}_2$-cutrank at least $r$. This implies a novel relation between rankwidth and a well-linkedness measure, defined entirely by balanced vertex cuts. As a byproduct, our result supports the notion of rank-expansion as a suitable candidate for measuring expansion in dense graphs.