Ends, tangles and critical vertex sets

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.