Paul D. Seymour

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.

Decomposition of regular matroids

Abstract It is proved that every regular matroid may be constructed by piecing together graphic and cographic matroids and copies of a certain 10-element matroid.

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.

The Complexity of Multiterminal Cuts

In the multiterminal cut problem one is given an edge-weighted graph and a subset of the vertices called terminals, and is asked for a minimum weight set of edges that separates each terminal from all the others. When the number

Graph minors. V. Excluding a planar graph

k

Graph minors: X. obstructions to tree-decomposition

of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. It is shown that the problem becomes NP-hard as soon as

Graph minors. III. Planar tree-width

k=3

Graph minors. I. Excluding a forest

, but can be solved in polynomial time for planar graphs for any fixed