Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

Plotkin, Rao, and Smith (SODA’97) showed that any graph with m edges and n vertices that excludes Kh as a depth O(l logn)-minor has a separator of size O(n/l + lh logn) and that such a separator can be found in O(mn/l) time. A time bound of O(m + n/l) for any constant ǫ > 0 was later given (W., FOCS’11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time O(poly(h)lm). This is a significant improvement for small h and l. If l = Ω(n ′ ) for an arbitrarily small chosen constant ǫ > 0, we get a time bound of O(poly(h)ln). The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on h) and running time O(poly(h)( √ ln + n/l)) when l = Ω(n ′ ). Our third algorithm has running time O(poly(h) √ ln) when l = Ω(n ′ ). It finds a separator of size O(n/l) + Õ(poly(h)l √ n) which is no worse than previous bounds when h is fixed and l = Õ(n). A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.