TY - GEN

T1 - Separator theorem for graphs with an excluded minor and its applications

AU - Alon, Noga

AU - Seymour, Paul

AU - Thomas, Robin

PY - 1990/1/1

Y1 - 1990/1/1

N2 - 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 h3/2n 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 2O(√n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure corresponds to G in time O(n3/2). We also describe a combinatorial application of our result which relates the tree-width of a graph to the maximum size of a Kh-minor in it.

AB - 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 h3/2n 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 2O(√n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure corresponds to G in time O(n3/2). We also describe a combinatorial application of our result which relates the tree-width of a graph to the maximum size of a Kh-minor in it.

UR - http://www.scopus.com/inward/record.url?scp=0024984223&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024984223&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0024984223

SN - 0897913612

T3 - Proc 22nd Annu ACM Symp Theory Comput

SP - 293

EP - 299

BT - Proc 22nd Annu ACM Symp Theory Comput

PB - Publ by ACM

T2 - Proceedings of the 22nd Annual ACM Symposium on Theory of Computing

Y2 - 14 May 1990 through 16 May 1990

ER -