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

2022

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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 from the well-known starcomb lemma for infinite graphs. Call a set U of vertices in a graph G tough in G if only finitely many components ofIn this fourth and final paper of the series, we structurally characterise the connected graphs G in which a given vertex set ⊆ U V G ( ) is tough. Our characterisations are phrased in terms of tree-decompositions, tangledistinguishing separators and tough subgraphs (a graph G is tough if its vertex set is tough in G). From the perspective of stars and combs, we thereby find structures whose existence is complementary to the existence of so-called undominating stars.

Section: And This Paper)mentioning

confidence: 99%

Duality theorems for stars and combs IV: Undominating stars

2022

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“…To formalise this, let us say that a subdivided star S dominates a comb C if infinitely many of the leaves of S are also teeth of C. A dominating star in a graph G then is a subdivided star S G ⊆ that dominates some comb C G ⊆ ; and a dominated comb in G is a comb C G ⊆ that is dominated by some subdivided star S G ⊆ . In the remaining three papers [1][2][3] of this series we shall find complementary structures to the existence of these substructures (again, with respect to some fixed set U ).…”

mentioning

confidence: 91%

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

2021

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“…In the second paper [2] of our series we determined structures whose existence is complementary to the existence of dominating stars or dominated combs-again in terms of normal trees or tree-decompositions.…”

Section: Introductionmentioning

confidence: 99%

Duality theorems for stars and combs III: Undominated combs

2021

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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.