(t, s)-Completely Independent Spanning Trees

In this paper we first define (t, s)-completely independent spanning trees, which is a generalization of completely independent spanning trees. A set of t spanning trees of a graph is (t, s)-completely independent if, for any pair of vertices u and v, among the set of t paths from u to v in the t spanning trees, at least s <= t paths are internally disjoint. By (t, s)-completely independent spanning trees, one can ensure any pair of vertices can communicate each other even if at most s - 1 vertices break down. We prove that every maximal planar graph has a set of (3, 2)-completely independent spanning trees, every tri-connected planar graph has a set of (3, 2)-completely independent spanning trees, and every 3D grid graph has a set of (3, 2)-completely independent spanning trees. Also one can compute them in linear time.