Extending the well-known star-comb lemma for infinite graphs, we characterise the graphs that do not contain an infinite comb or an infinite star, respectively, attached to a given set of vertices. We offer several characterisations: in terms of normal trees, tree-decompositions, ranks of rayless graphs and tangle-distinguishing separators.

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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 false|Gfalse|Γ is a quotient of Diestel's (denoted by false|Gfalse|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 false|Gfalse|Θ and our construction of false|Gfalse|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 false|Gfalse|Θ is the finest such compactification, and our false|Gfalse|Γ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.

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