English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show that an arbitrary infinite graph $G$ can be compactified by its endsplus its critical vertex sets, where a finite set $X$ of vertices of aninfinite graph is critical if its deletion leaves some infinitely manycomponents each with neighbourhood precisely equal to $X$.  We further provide a concrete separation system whose $\aleph_0$-tangles areprecisely the ends plus critical vertex sets. Our tangle compactification$\vert G\vert_{\Gamma}$ is a quotient of Diestel's (denoted by $\vertG\vert_{\Theta}$), and both use tangles to compactify a graph in much the sameway as the ends of a locally finite and connected graph compactify it in itsFreudenthal compactification.  Finally, generalising both Diestel's construction of $\vert G\vert_{\Theta}$and our construction of $\vert G\vert_{\Gamma}$, we show that $G$ can becompactified by every inverse limit of compactifications of the sets ofcomponents obtained by deleting a finite set of vertices. Diestel's $\vertG\vert_{\Theta}$ is the finest such compactification, and our $\vertG\vert_{\Gamma}$ is the coarsest one. Both coincide if and only if all tanglesare ends. This answers two questions of Diestel.

Ends, tangles and critical vertex sets