is known that the problem of determining, given a planar graph C: and an integer m; whether there exists a congestion-1 embedding of G' into an m x k-grid is NP-compiete for a fixed integer k greater than or equal to 3. It is also known that the problem for k = 3 is NP-complete even if G is restricted to an acyclic graph. The complexity of the problem for ic = 2 was left open. In this paper, we show that for k = 2, the problem can be solved in polynomial time if G is restricted to a tree, while the problem is NP-complete even if G is restricted to an acyclic graph.