Completely independent spanning trees in maximal planar graphs

Let G be a graph. Let T-1, T-2,..., T-k be spanning trees in G. If for any two vertices u, v in G, the paths from u to v in, T-1, T-2,..., T-k are pairwise openly disjoint, then we say that T-1, T-2,..., T-k are completely independent spanning trees in G. In this paper, we show that there are two completely independent spanning trees in any 4-connected maximal planar graph. Our proof induces a linear-time algorithm for finding such trees. Besides, we show that given a graph G, the problem of deciding whether there exist two completely independent spanning trees in G is NP-complete.