Bollobás and Thomason showed that every 22k-connected graph is k-linked. Their result used a dense graph minor. In this paper we investigate the ties between small graph minors and linkages. In particular, we show that a 6-connected graph with a K− 9 minor is 3-linked. Further, we show that a 7-connected graph with a K− 9 minor is (2, 5)-linked. Finally, we show that a graph of order n and size...

It is well known that any planar graph contains at most O(n) complete subgraphs. We extend this to an exact characterization: G occurs O(n) times as a subgraph of any planar graph, if and only if G is three-connected. We generalize these results to similarly characterize certain other minor-closed families of graphs; in particular, G occurs O(n) times as a subgraph of the Kb,c-free graphs, b ≥ ...

We show that the countably infinite union of infinite grids, H say, is minor-universal in the class of all graphs that can be drawn in the plane without vertex accumulation points, i.e. that H contains every such graph as a minor. Furthermore, we characterize the graphs that occur as minors of the infinite grid by a natural topological condition on their embeddings.

In 1981, Seymour proved a conjecture of Welsh that, in a connected matroid M , the sum of the maximum number of disjoint circuits and the minimum number of circuits needed to cover M is at most r∗(M) + 1. This paper considers the set Ce(M) of circuits through a fixed element e such that M/e is connected. Let νe(M) be the maximum size of a subset of Ce(M) in which any two distinct members meet o...

A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that nd tree-decompositions, algorithms that use tree-decompositions to solve hard problems eeciently, graph minor theory, and some applications. The paper contains an extensive bibliography.

This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems.

The Lescure–Meyniel conjecture is the analogue of Hadwiger’s for immersion order. It states that every graph G contains complete Kχ(G) as an immersion, and like its minor-order counterpart it open even graphs with independence number 2. We show α(G)≥2 no hole length between 4 2α(G) satisfies this conjecture. In particular, C4-free α(G)=2 give another generalisation corollary, follows. Let H be ...