On the vertex arboricity of planar graphs of diameter two Original Research Article

An induced forest k-partition of a graph G is a k-partition image of image such that each image, image, is a forest. The vertex arboricity of a graph G is the minimum positive integer k for which G has an induced forest k-partition. In this paper, we show that the vertex arboricity of planar graphs of diameter 2 is no more than two, and the induced forest 2-partition problem is NP-complete for graphs of diameter 2.