Separator theorem for graphs with an excluded minor and its applications

Let G be an n-vertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given h-vertex graph H. We prove that there is a set X of no more than h^3/2n ^1/2 vertices of G whose deletion creates a graph in which the total weight of every connected component is at most 1/2 . This extends significantly a well-known theorem of R.J. Lipton and R.E. Tarjan for planar graphs. We exhibit an algorithm which finds, given an n-vertex graph G with weights as above and an h-vertex graph H, either such a set X or a minor of G isomorphic to H. The algorithm runs in time O(h ^1/2 n ^1/2 m), where m is the number of edges of G plus the number of its vertices. Our results supply extensions of the many known applications of the Lipton-Tarjan separator theorem from the class of planar graphs (or that of graphs with bounded genus) to any class of graphs with an excluded minor. For example, it follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/√log n in polynomial time, find that size exactly and find the chromatic number of G in time 2^O(√n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure corresponds to G in time O(n^3/2). We also describe a combinatorial application of our result which relates the tree-width of a graph to the maximum size of a K_h-minor in it.