Canonical graph decompositions via coverings
Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Jan Kurkofka
TL;DR
The paper develops a canonical, r-dependent decomposition of finite graphs into locally highly connected parts whose global arrangement is captured by a finite model graph H=H(G,r). This is achieved by passing to the r-local cover G_r, applying a canonical, tangle-based tree-decomposition on G_r, and projecting the resulting structure back to G via the orbit space T_r/𝒟_r, yielding a canonical H-decomposition. The framework extends to locally finite quasi-transitive graphs and to Cayley graphs, providing a canonical decomposition of finitely generated groups into local parts whose global organization is displayed by a finite graph. By integrating covering-space ideas with graph tangles, the work offers a principled local/global lens for graph and group structure, with potential applications to structural theory and algorithmic decomposition methods.
Abstract
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 $\it model$. 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 while unfolding its global structure. In this way, the tangle theory from graph minors is brought to bear canonically on arbitrary graphs, which need not be tree-like. Our theorem extends to locally finite quasi-transitive graphs, and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition for finitely generated groups into local parts, whose relative structure is displayed by a graph.