On the vertex-arboricity of planar graphs

The vertex-arboricity a(G)a(G) of a graph GG is the minimum number of subsets into which the set of vertices of GG can be partitioned so that each subset induces a forest. It is well-known that a(G)≤3a(G)≤3 for any planar graph GG. In this paper we prove that a(G)≤2a(G)≤2 whenever GG is planar and either GG has no 4-cycles or any two triangles of GG are at distance at least 3.