# The average cut-rank of graphs

## TL;DR

This paper introduces the average cut-rank as a new graph parameter, explores its properties under vertex-minors, and characterizes graphs with bounded average cut-rank, including explicit classifications up to 3/2.

## Contribution

It defines the average cut-rank, proves its invariance under vertex-minors, and characterizes classes of graphs with bounded average cut-rank, including explicit classifications up to 3/2.

## Key findings

- Average cut-rank does not increase under vertex-minors.
- Graphs with bounded average cut-rank are characterized by bounded neighborhood diversity.
- Explicit classification of graphs with average cut-rank at most 3/2.

## Abstract

The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $\alpha$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $\alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $\alpha$ for each real $\alpha\ge0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph.