Duality theorems for stars and combs II: Dominating stars and dominated combs

Abstract

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.

End‐faithful spanning trees in graphs without normal spanning trees

“…In this section, we prove that for every normally traceable graph $$G$$, having a rayless spanning tree is equivalent to all the ends of $$G$$ being dominated. Our proof builds on the following theorem from the third paper of the star‐comb series [2–5] that hides a six page argument:…”

Duality theorems for stars and combs I: Arbitrary stars and combs

“…The remaining ‘moreover’ part is a consequence of [1, Theorem 1], which is why its proof is placed in the second paper of our series, cf. [1, Section 2].…”

“…The remaining ‘moreover’ part is a consequence of [1, Theorem 1], which is why its proof is placed in the second paper of our series, cf. [1, Section 2]. To see immediately that a locally finite normal tree as in (ii) is more specific than a comb when is infinite, apply Lemma 2.3 to .…”

“…A dominating star in a graph then is a subdivided star that dominates some comb ; and a dominated comb in is a comb that is dominated by some subdivided star . In the remaining three papers [1–3] of this series we shall find complementary structures to the existence of these substructures (again, with respect to some fixed set ).…”

A representation theorem for end spaces of infinite graphs

“…For a concept that is somewhat stronger than our notion of envelopes (but whose existence proof is significantly more involved) see Polat's multiendings from [41]; a precursor to our notion of envelopes has been used in Bürger and the first author's proof of [8,Theorem 1].…”