Connectivity and tree structure in finite graphs

TL;DR

This paper investigates how certain separation systems in finite graphs can be organized into nested, automorphism-invariant structures, leading to tree-decompositions that distinguish maximal inseparable vertex sets called $k$-blocks.

Contribution

It extends existing theorems by providing conditions for automorphism-invariant nested separation systems and constructing comprehensive tree-decompositions for all $k$-blocks simultaneously.

Findings

Existence of automorphism-invariant nested separation systems

Tree-decompositions that distinguish $k$-blocks in finite graphs

Extension of Tutte's theorem for $k=2$ to general $k$

Abstract

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the $k$-blocks -- the maximal vertex sets that cannot be separated by at most $k$ vertices -- of a graph $G$ live in distinct parts of a suitable tree-decomposition of $G$ of adhesion at most $k$, whose decomposition tree is invariant under the automorphisms of $G$. This extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a similar theorem of Tutte for $k=2$. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all $k$ simultaneously, all the $k$-blocks of a finite graph.