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Rankwidth of Graphs with Balanced Separations: Expansion for Dense Graphs

Emile Anand

TL;DR

This work establishes a tight connection between rankwidth and a robust balanced-cutrank measure for graphs, showing that high rankwidth implies the existence of large induced subgraphs that resist low-rank balanced separations. The authors develop a rank-tangle framework and a compression argument to extract a subgraph whose minimum balanced cutrank is large, paralleling grid-minor style certificates for sparse graphs. They prove the inequality $\frac{1}{72}\mathrm{rankwidth}(G) \le \max_{H\subseteq G} \operatorname{min-bal-cutrank}(H) \le \mathrm{rankwidth}(G)$, supporting rank-expansion as a meaningful dense-graph expansion analogue and suggesting new algorithmic certifiation directions. The results bridge linear-algebraic expansion concepts with structural graph theory and open avenues for dense-graph testing and approximation via rankwidth-inspired certificates.

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.

Rankwidth of Graphs with Balanced Separations: Expansion for Dense Graphs

TL;DR

This work establishes a tight connection between rankwidth and a robust balanced-cutrank measure for graphs, showing that high rankwidth implies the existence of large induced subgraphs that resist low-rank balanced separations. The authors develop a rank-tangle framework and a compression argument to extract a subgraph whose minimum balanced cutrank is large, paralleling grid-minor style certificates for sparse graphs. They prove the inequality , supporting rank-expansion as a meaningful dense-graph expansion analogue and suggesting new algorithmic certifiation directions. The results bridge linear-algebraic expansion concepts with structural graph theory and open avenues for dense-graph testing and approximation via rankwidth-inspired certificates.

Abstract

We prove that every graph of rankwidth at least contains an induced subgraph whose minimum balanced cutrank is at least , which implies a vertex subset where every balanced separation has -cutrank at least . 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.

Paper Structure

This paper contains 8 sections, 10 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Proof overview.
  3. Preliminaries
  4. Graphs and matrices.
  5. Remarks on Neighborhood Equivalence Classes
  6. Proof approach
  7. The Maximal Well-Behaved Separation
  8. The Compressed Subgraph $H$

Key Result

Theorem 1.1

Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.

Theorems & Definitions (28)

  • Definition 1.1: Rank-decomposition and rankwidth Oum_Seymour_2006Voigt
  • Definition 1.2: Rank-Expansion
  • Theorem 1.1
  • Definition 2.1: Cutrank
  • Definition 2.2: Balanced partition
  • Definition 2.3: Minimum Balanced Cutrank of Graph $G$
  • Definition 2.4: Rank-tangle of order $k$
  • Definition 2.5: Well-behaved separations
  • Theorem 2.1
  • Definition 2.6: Tree Decomposition of Graph
  • ...and 18 more