Partitionability to two trees is NP-complete

We show that P2T-the problem of deciding whether the edge set of a simple graph can be partitioned into two trees or not-is NP-complete. It is a well known that deciding whether the edge set of a graph can be partitioned into k spanning trees or not is in P [1]. Recently András Frank asked what we know about partitioning the edge set of a graph into k (not necessarily spanning) trees. One can easily see that whether a simple graph is a tree or not is in P. It was shown by Király that the problem of deciding if the edge set of a simple graph is the disjoint union of three trees is NP-complete by reducing the 3-colorability problem to it [3]. Now we prove that for two trees the problem is also NP-complete. First we define the problem precisely.