INDEPENDENT TREES IN GRAPHS

If G is a finite undirected graph and s is a vertex of G, then two spanning trees T1 and T2 in G are called s - independent if for each vertex x in G the paths from x to s in T1 and T2 are openly disjoint. It is known that the following statement is true for k less-than-or-equal-to 3: If G is k - connected, then there are k pairwise s - independent spanning trees in G. As a main result we show that this statement is also true for k = 4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weakly s - independent spanning trees (i.e., the tree paths from a vertex to s are edge disjoint) and for directed graphs.