Duality theorems for stars and combs IV: 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 from the well-known star—comb lemma for infinite graphs. Call a set U of vertices in a graph G tough in G if only finitely many components of G − X meet U for every finite vertex set X ⊆ V (G). In 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, tangle-distinguishing 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.