On the vertex partition of planar graphs into forests with bounded degree Abstract   Let     G    (  V  ,  E  )     be a graph. A     (    xcex94    d  1    ,    xcex94    d  2    )    partition of a graph G is the partition of V(G) into two nonempty subsets V1 and V2, such that GV1 and GV2 are graphs with maximum degree at most d1 and d2, respectively. A similar definition can be given for the notation     (   F   d  1    ,   F   d  2    )    partition if GV1 and GV2 are forests with maximum degree at most d1 and d2, respectively.  For any d1 and d2, Montassier and Ochem constructed graphs with girth 4 which do not admit     (    xcex94    d  1    ,    xcex94    d  2    )    partition. In this paper, we show that planar graphs with girth at least 5 and without adjacent 5cycles admit     (   F  3   ,   F  3   )    partition.

Partitioning Cographs into Two Forests and One Independent Set We consider a variation of arboricity, where a graph is partitioned into p forests and q independent sets. These problems are NPcomplete in general, but polynomialtime solvable in the class of cographs; in fact, for each p and q there are only finitely many minimal nonpartitionable cographs. In previous investigations it was revealed that when (p0) or (p1), these minimal nonpartitionable cographs can be uniformly described as one family of obstructions valid for all values of q. We investigate the next case, when (p2); we provide the complete family of minimal obstructions for (p2, q1), and find that they include more than just the natural extensions of the previously described obstructions for (p2, q0). Thus a uniform description for all q seems unlikely already in the case (p2).