# On the block number of graphs

## TL;DR

This paper investigates the structure of graphs based on their block number, providing a decomposition theorem for graphs without large blocks, and establishing bounds on block numbers in minor-closed classes and relation to tree-width.

## Contribution

It introduces a structure theorem for graphs without a (k+1)-block, linking block number to tree-decomposition and degree constraints, and bounds block numbers in minor-closed classes.

## Key findings

- Graphs without (k+1)-blocks have a specific tree-decomposition structure.
- Block number in minor-closed classes is bounded by a cube root of the graph size.
- High tree-width guarantees the existence of a minor with a k-block.

## Abstract

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a $k$-block. We prove a structure theorem for graphs without a $(k+1)$-block, showing that every such graph has a tree-decomposition in which every torso has at most $k$ vertices of degree $2k^2$ or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most $2k^2$. We also study $k$-blocks in graphs from classes of graphs $\mathcal{G}$ that exclude some fixed graph as a topological minor, and prove that every $G \in \mathcal{G}$ satisfies $\beta(G) \leq c\sqrt[3]{|G|}$ for some constant $c = c( \mathcal{G})$. Moreover, we show that every graph of tree-width at least $2k^2$ has a minor containing a $k$-block. This bound is best possible up to a multiplicative constant.