Canonical graph decompositions and local separations: From infinite coverings to a finite combinatorial theory

Abstract: Every finite graph $G$ can be decomposed in a canonical way that displays its local connectivity-structure [DJKK22]. For this, they consider a suitable covering of $G$, an inherently infinite concept from Topology, and project its tangle-tree structure to $G$. We present a construction of these decompositions via a finite combinatorial theory of local separations, which we introduce here and which is compatible with the covering-perspective.