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.

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs from the well-known starcomb lemma for infinite graphs. Call a set U of vertices in a graph G tough in G if only finitely many components ofIn this fourth and final paper of the series, we structurally characterise the connected graphs G in which a given vertex set ⊆ U V G ( ) is tough. Our characterisations are phrased in terms of tree-decompositions, tangledistinguishing separators and tough subgraphs (a graph G is tough if its vertex set is tough in G). From the perspective of stars and combs, we thereby find structures whose existence is complementary to the existence of so-called undominating stars.

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the second paper of the series, we present duality theorems for combinations of stars and combs: dominating stars and dominated combs. As dominating stars exist if and only if dominated combs do, the structures complementary to them coincide. Like for arbitrary stars and combs, our duality theorems for dominated combs (and dominating stars) are phrased in terms of normal trees or tree-decompositions. The complementary structures we provide for dominated combs unify those for stars and combs and allow us to derive our duality theorems for stars and combs from those for dominated combs. This is surprising given that our complementary structures for stars and combs are quite different: Those for stars are locally finite whereas those for combs are rayless.

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the third paper of the series, we present duality theorems for a combination of stars and combs: undominated combs. We describe their complementary structures in terms of rayless trees and of tree-decompositions. Applications include a complete characterisation, in terms of normal spanning trees, of the graphs whose rays are dominated but which have no rayless spanning tree. Only two such graphs had so far been constructed, by Seymour and Thomas and by Thomassen. As a corollary, we show that graphs with a normal spanning tree have a rayless spanning tree if and only if all their rays are dominated.

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