A Separator Theorem for Graphs with an Excluded Minor and its Applications

ions (Extended Abstract) Noga Alon Paul Seymour Robin Thomas 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 h3/2 n1/2 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 Lipton and 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(h1/2 n1/2 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 ...

• Type: Conference
• Year: 1990
• Where: STOC
• Authors: Noga Alon, Paul D. Seymour, Robin Thomas
• Keywords: Algorithms, H-vertex Graph H, N-vertex Graph, Planar Graphs, STOC 1990