Duality theorems for stars and combs III: Undominated 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 third paper of the series, we present duality theorems for a combination of stars and combs: undominated combs. We describe their complementary structures in terms of rayless trees and of tree-decompositions. Applications include a complete characterisation, in terms of normal spanning trees, of the graphs whose rays are dominated but which have no rayless spanning tree. Only two such graphs had so far been constructed, by Seymour and Thomas and by Thomassen. As a corollary, we show that graphs with a normal spanning tree have a rayless spanning tree if and only if all their rays are dominated.

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:…”

“…An end of $$G$$ is dominated if one (equivalently: each) of its rays is dominated, see [6]. For a connected graph $$G$$, having a rayless spanning tree is equivalent to all the ends of $$G$$ being dominated if $$G$$ is normally spanned [4] or if $$G$$ does not contain a subdivision of $${T}_{{\aleph }_{1}}$$ [12]. Our second main result extends these results, and any $${T}_{{\aleph }_{1}}$$ with all tops witnesses that this extension is proper.…”

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

“…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 ).…”

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

“…In a series of four papers we determine structures whose existence is complementary to the existence of two substructures that are particularly fundamental to the study of connectedness in infinite graphs: stars and combs. See [3] for a comprehensive introduction, and a brief overview of results, for the entire series of four papers ([3,1,2] and this paper).…”