A Separator Theorem in Minor-Closed Classes

It is shown that for each t, there is a separator of size O(t√n) in any n-vertex graph G with no K_t-minor. This settles a conjecture of Alon, Seymour and Thomas (J. Amer. Math. Soc, 1990 and STOC´90), and generalizes a result of Djidjev (1981), and Gilbert, Hutchinson and Tarjan (J. Algorithm, 1984), independently, who proved that every graph with n vertices and genus g has a separator of order O(√gn), because K_t has genus Ω(t²). The bound O(t√n) is best possible because every 3-regular expander graph with n vertices is a graph with no K_t-minor for t = cn^1/2, and with no separator of size dn for appropriately chosen positive constants c, d. In addition, we give an O(n²) time algorithm to obtain such a separator, and then give a sketch how to obtain such a separator in O(n^1+ε) time for any ε > 0. Finally, we discuss several algorithm aspects of our separator theorem, including a possibility to obtain a separator of order g(t)√n, for some function g of t, in an n-vertex graph G with no K_t-minor in O(n) time.