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 complementary, to stars or combs from well-known star—comb lemma for infinite graphs. Call set U vertices graph G tough if only finitely many components ? X meet every finite vertex ? V ( ) . this fourth and final paper series, structurally characterise connected graphs which given tough. Our characterisations are phrased terms tree-decompositions, tangle-distinguishing separators subgraphs (a its G). From perspective combs, thereby find complementary so-called undominating stars.