Graph minors and linkages

Extremal Functions for Graph Linkages and Rooted Minors

Graph minors. XXI. Graphs with unique linkages

A linkage L in a graph G is a subgraph each component of which is a path, and it is vital if V (L) = V (G) and there is no other linkage in G joining the same pairs of vertices. We show that, if G has a vital linkage with p components, then G has tree-width bounded above by a function of p. This is the major step in the proof of the unproved lemma from Graph Minors XIII, and it has a number of ...

The excluded minor structure theorem, and linkages in large graphs of bounded tree-width

At the core of the Robertson-Seymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some particular aspects needed at a corresponding stage of the proof of the main result of their theory, the ...

Graph minors and graphs on surfaces

Graph minors and the theory of graphs embedded in surfaces are fundamentally interconnected. Robertson and Seymour used graph minors to prove a generalization of the Kuratowski Theorem to arbitrary surfaces [37], while they also need surface embeddings in their Excluded Minor Theorem [45]. Various recent results related to graph minors and graphs on surfaces are presented.

Hypertree-depth and minors in hypergraphs

We introduce two new notions for hypergraphs, hypertree-depth and minors in hypergraphs. We characterise hypergraphs of bounded hypertree-depth by the ultramonotone robber and marshals game, by vertex-hyperrankings and by centred hypercolourings. Furthermore, we show that minors in hypergraphs are ‘well-behaved’ with respect to hypertree-depth and other hypergraph invariants, such as generalise...