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… Expand

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.Expand

It is shown that the problem becomes NP-hard as soon as $k=3$, but can be solved in polynomial time for planar graphs for any fixed $k$, if the planar problem is NP- hard, however, if £k$ is not fixed.Expand

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.

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… Expand

A polynomial-time algorithm to approximate the branch-width of certain symmetric sub-modular functions, and gives two applications to graph "clique-width" and the area of matroid branch- width.Expand

A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of… Expand

The obstructions to the existence of a graph are studied to find a minimax formula relating tree-width with the largest such obstructions and a “tree-decomposition” of the graph into pieces corresponding with the obstructions.Expand