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  • Print publication year: 2015
  • Online publication date: July 2015

7 - New tools and results in graph minor structure theory

Summary

Abstract

Graph minor theory of Robertson and Seymour is a far reaching generalization of the classical Kuratowski–Wagner theorem, which characterizes planar graphs in terms of forbidden minors. We survey new structural tools and results in the theory, concentrating on the structure of large t-connected graphs, which do not contain the complete graph Kt as a minor.

1 Introduction

Graphs in this paper are finite and simple, unless specified otherwise. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Numerous theorems in structural graph theory describe classes of graphs which do not contain a fixed graph or a collection of graphs as a minor. A classical example of such a description is the Kuratowski–Wagner theorem [92,93].

Theorem 1.1A graph is planar if and only if it does not contain K5 or K3,3 as a minor.

(We will say that G contains H as a minor, if H is isomorphic to a minor of G, and we will use the notation HG to denote this. The notation is justified as the minor containment is, indeed, a partial order. We say that G is H-minor free if G does not contain H as a minor.)

Clearly a graph is a forest if and only if it does not contain K3 as a minor. In [16] Dirac proved that a graph does not contain K4 as a minor if and only if it is series-parallel. In [93] Wagner characterizes graphs which do not contain K5 as a minor, as follows.

Theorem 1.2A graph does not contain K5 as a minor if and only if it can be obtained by 0-, 1 and 2 and 3-clique sum operations from planar graphs and V8. (The graph V8 is shown on Figure 1.)

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Surveys in Combinatorics 2015
  • Online ISBN: 9781316106853
  • Book DOI: https://doi.org/10.1017/CBO9781316106853
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