Ends, tangles and critical vertex sets. Issue 9 (10th May 2019) - Kurkofka, Jan; Pitz, Max
Abstract: We show that an arbitrary infinite graph G can be compactified by its ends plus its critical vertex sets, where a finite set X of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to X . We further provide a concrete separation system whose ℵ0 ‐tangles are precisely the ends plus critical vertex sets. Our tangle compactification | G | Γ is a quotient of Diestel's (denoted by | G | Θ ), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of | G | Θ and our construction of | G | Γ, we show that G can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's | G | Θ is the finest such compactification, and our | G | Γ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.