 
Summary: A Separator Theorem for Graphs with an Excluded Minor and its
Applications
Noga Alon
IBM Almaden Research Center, San Jose, CA 95120 ,USA and
Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
Paul Seymour
BellCore, 445 South St., Morristown, NJ 07960, USA
Robin Thomas
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Abstract
Let G be an nvertex graph with nonnegative weights
whose sum is 1 assigned to its vertices, and with no
minor isomorphic to a given hvertex 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 compo
nent is at most 1/2. This extends significantly a
wellknown theorem of Lipton and Tarjan for pla
nar graphs. We exhibit an algorithm which finds,
