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.
Original language | English |
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Pages (from-to) | 127-139 |
Number of pages | 13 |
Journal | Journal of Graph Theory |
Volume | 100 |
Issue number | 1 |
Early online date | 8 Dec 2021 |
DOIs | |
Publication status | Published - May 2022 |
Keywords
- complementary
- dual
- duality
- infinite graph
- normal tree
- rayless spanning tree
- star–comb lemma
- star-decomposition
- tree-decomposition
- undominated comb
- undominated ends