Duality theorems for stars and combs IV: Undominating stars

Abstract

AbstractIn 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 of vertices in a graph tough in if only finitely many components of meet for every finite vertex set . In this fourth and final paper of the series, we structurally characterise the connected graphs in which a given vertex set is tough. Our characterisations are phrased in terms of tree‐decompositions, tangle‐distinguishing separators and tough subgraphs (a graph is tough if its vertex set is tough in ). From the perspective of stars and combs, we thereby find structures whose existence is complementary to the existence of so‐called undominating stars.