Neil Robertson

Graph Minors

Abstract The main result of this series serves to reduce several problems about general graphs to problems about graphs which can “almost” be drawn in surfaces of bounded genus. In applications of the theorem we usually need to encode such a nearly embedded graph as a hypergraph which can be drawn completely in the surface. The purpose of this paper is to show how to “tidy up” near-embeddings to facilitate the encoding procedure.

Graph minors. II. Algorithmic aspects of tree-width

We introduce an invariant of graphs called the tree-width, and use it to obtain a polynomially bounded algorithm to test if a graph has a subgraph contractible to H, where H is any fixed planar graph. We also nonconstructively prove the existence of a polynomial algorithm to test if a graph has tree-width ≤ w, for fixed w. Neither of these is a practical algorithm, as the exponents of the polynomials are large. Both algorithms are derived from a polynomial algorithm for the DISJOINT CONNECTING PATHS problem (with the number of paths fixed), for graphs of bounded tree-width.

Graph minors. XIII: the disjoint paths problem

Abstract We describe an algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G , decide if there are k mutually vertex-disjoint paths of G joining the pairs.

Graph Minors. XX. Wagner's conjecture

We prove Wagners conjecture, that for every infinite set of finite graphs, one of its members is isomorphic to a minor of another.

Graph minors. V. Excluding a planar graph

We prove that for every planar graph H there is a number w such that every graph with no minor isomorphic to H can be constructed from graphs with at most w vertices, by piecing them together in a tree structure. This has several consequences; for example, it implies that: (i) if A is a set of graphs such that no member is isomorphic to a minor of another, and some member of A is planar, then A is finite; (ii) for every fixed planar graph H there is a polynomial time algorithm to test if an arbitrary graph has a minor isomorphic to H; (iii) there is a generalization of a theorem of Erdos and Posa (concerning the maximum number of disjoint circuits in a graph) to planar structures other than circuits.

Graph minors: X. obstructions to tree-decomposition

Abstract Roughly, a graph has small “tree-width” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: 1. (i) a minimax formula relating tree-width with the largest such obstructions 2. (ii) an association between such obstructions and large grid minors of the graph 3. (iii) a “tree-decomposition” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers.

Graph minors. III. Planar tree-width

Abstract The “tree-width” of a graph is defined and it is proved that for any fixed planar graph H, every planar graph with sufficiently large tree-width has a minor isomorphic to H. This result has several applications which are described in other papers in this series.

Graph minors. I. Excluding a forest

The path-width of a graph is the minimum value ofk such that the graph can be obtained from a sequence of graphsG1,…,Gr each of which has at mostk + 1 vertices, by identifying some vertices ofGi pairwise with some ofGi+1 (1 ≤ i < r). For every forestH it is proved that there is a numberk such that every graph with no minor isomorphic toH has path-width≆k. This, together with results of other papers, yields a “good” algorithm to test for the presence of any fixed forest as a minor, and implies that ifP is any property of graphs such that some forest does not have propertyP, then the set of minor-minimal graphs without propertyP is finite.

The Four-Colour Theorem

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Hakens in several respects.

Quickly excluding a planar graph

Abstract In an earlier paper, the first two authors proved that for any planar graph H , every graph with no minor isomorphic to H has bounded tree width; but the bound given there was enormous. Here we prove a much better bound. We also improve the best known bound on the tree-width of planar graphs with no minor isomorphic to a g × g grid.