Graph Minors

Graph Minors

For a given graph G and integers b, f ≥ 0, let S be a subset of vertices of G of size b + 1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most n/ b+f b such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several e...

Treewidth and graph minors

We shall touch upon the theory of Graph Minors by Robertson and Seymour. This theory gives a very general condition under which a graph problem has a polynomial time algorithm (though the algorithms that come out of the theory are often not practical). We shall review a small part of this theory, and illustrate how dynamic programming can be used to solve some NP-hard problems on restricted cla...

Graph minors and linkages

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...

New Proofs in Graph Minors

Graph Minors. XX. Wagner's conjecture

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