On May 18, 2022, Jan Kurkofka from the University of Birmingham gave an online talk at the Virtual Discrete Math Colloquium on the canonical decomposition of finite graphs into highly connected local parts. The title of his talk was “Canonical Graph Decompositions via Coverings“.
We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser model. This is a simpler graph determined entirely by the decomposition, not imposed.
The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure at the intended level of locality while unfolding its global structure.
Our theorem extends to locally finite quasi-transitive graphs and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition theorem for finitely generated groups into local parts, whose relative structure is displayed by a graph.
Joint work with Reinhard Diestel, Raphael W. Jacobs and Paul Knappe.