Vertex Cuts

Abstract

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2-connected graphs, that are not 3-connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2-connected graphs to k-connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k-CS-transitive graphs. C 2014Wiley Periodicals, Inc. J. Graph Theory 80: 2015

The Planar Cubic Cayley Graphs

“…Thus in both cases we obtained a contradiction to the maximality of P , which proves our claim (11).…”

Canonical tree-decompositions of finite graphs I. Existence and algorithms

“…Extending results of Tutte [9] and of Dunwoody and Krön [5], three of us and Maya Stein showed that every finite graph G admits, for every integer k, a tree-decomposition (T, V) of adhesion < k that distinguishes all its k-blocks [3]. These decompositions are canonical in that the map G → (T, V) commutes with graph isomorphisms.…”

“…Indeed, as (A, V ) ≤ (V, A), a consistent set of separations containing (V, A) must not contain the inverse of (A, V ), which is (V, A). 5 Note that consistent sets of separations can contain improper separations of the form (A, V ).…”

Structure Trees, Networks and Almost Invariant Sets

“…These are connected graphs that have more than one infinite component after removing finitely many vertices. We were able to develop such a theory in [14]. In the course of our work on this, we realised that we could develop a theory of structure trees for finite graphs that generalised the theory of Tutte [33], who obtained a structure tree result for 2-connected finite graphs that are not 3-connected.…”