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

The tree-width of a graph G is the minimum k such that G may be decomposed into a "tree-structure" of pieces each with at most k + l vertices. We prove that this equals the maximum k such that there… Expand