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 ${\mathrm{\aleph }}_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\mathrm{\mid }G{\mathrm{\mid }}_{\mathrm{\Gamma }}$ is a quotient of Diestel’s (denoted by $\vert G\vert_{\Theta}$), 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 $\mathrm{\mid }G{\mathrm{\mid }}_{\mathrm{\Theta }}$ and our construction of $\mathrm{\mid }G{\mathrm{\mid }}_{\mathrm{\Gamma }}$, 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 $\vert G\vert_{\Theta}$isthefinestsuchcompactification,andour$\vert G\vert_{\Gamma}$ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.