Abstract. We survey a number of results about minors of graphs which we have recently obtained. They are basically of three types:
(i) results concerning the structure of the graphs with no minor isomorphic to a fixed graph
(ii) results concerning a conjecture of K. Wagner. that for any infinite set of graphs one of its members is isomorphic to a minor of another. and
(iii) algorithmic results concerning the DISJOINT CONNECTING PATHS problem.
There are two fundamental questions which motivate the work we report on here.
(A) (K. Wagner's well-quasi-ordering conjecture). Is it true that for every infinite sequence G1, G2, … of graphs, there exist i, j with i < j such that Gi is isomorphic to a minor of Gj?
[Graphs in this paper are finite and may have loops or multiple edges. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges.]
(B) (The DISJOINT CONNECTING PATHS problem). If k ≥ 0, is there a polynomially-bounded algorithm to decide, given a graph G and vertices s1, …, sk, t1, …, tk of G, whether there are k mutually disjoint paths P1, …, Pk of G where Pi has ends si, ti (1 ≤ i ≤ k)?
[Two paths are disjoint if they have no common vertices.] Some of the background to these questions is discussed in sections 2 and 3.