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 Gtough 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. [ABSTRACT FROM AUTHOR]