# Some Recent Progress and Applications in Graph Minor Theory

@article{Kawarabayashi2007SomeRP, title={Some Recent Progress and Applications in Graph Minor Theory}, author={Ken-ichi Kawarabayashi and Bojan Mohar}, journal={Graphs and Combinatorics}, year={2007}, volume={23}, pages={1-46} }

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough'' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed.

## 57 Citations

### Complete graph minors and the graph minor structure theorem

- MathematicsJ. Comb. Theory, Ser. B
- 2013

### Proof theory of graph minors and tree embeddings

- Mathematics, Computer Science
- 2018

This thesis establishes a narrow corridor for the possible proof-theoretic strength of many strong combinatorial principles, including the graph minor theorem, immersion theorem, theorems about patchwork containment, and various restrictions, extensions and labelled versions of these theorem.

### Forbidden Minors: Finding the Finite Few

- Mathematics
- 2017

The Graph Minor Theorem of Robertson and Seymour associates, to any graph property whatsoever, a finite, characteristic list of graphs. We view this as an impressive generalization of Kuratowski’s…

### Graph Minors

- Mathematics
- 2009

For a given graph G and integers b, f ≥ 0, let S be a subset of vertices of G of size b + 1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by…

### Excluded-Minor Characterization of Apex-Outerplanar Graphs

- MathematicsGraphs Comb.
- 2016

The class of outerplanar graphs is minor-closed and can be characterized by two excluded minors: $$K_4$$K4 and $$K_{2,3}$$K2,3. The class of graphs that contain a vertex whose removal leaves an…

### Parameters Tied to Treewidth

- Computer Science, MathematicsJ. Graph Theory
- 2017

Several graph parameters tied to treewidth are surveyed, including separation number, tangle number, well‐linked number, and Cartesian tree product number, which improve known bounds, provide simpler proofs, and show that the inequalities presented are tight.

### On Treewidth and Graph Minors

- Mathematics, Computer Science
- 2014

This thesis determines exactly the treewidth of the line graph of a complete graph, up to lower order terms in general, and exactly whenever the complete multipartite graph is regular, and generalises a result by Lucena.

### Directed Graph Embeddings

- MathematicsArXiv
- 2017

The usefulness of the directed embedding operation is demonstrated be characterizing all TDAGs with parallel-width at most $k$, which generalizes earlier characterizations of series-parallel graphs.

### Connectivity, tree-decompositions and unavoidable-minors

- Mathematics
- 2015

The results in this thesis are steps toward bridging the gap between the handful of exact structure theorems known for minor-closed classes of graphs, and the very general, yet wildly qualitative,…

### Clique Minors in Cartesian Products of Graphs

- MathematicsArXiv
- 2007

A rough structural characterisation theorem for Cartesian products with bounded Hadwiger number implies that if the product of two sufficiently large graphs has bounded HadWiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindrical grid, or - a toroidal grid.

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