On the vertex arboricity of planar graphs of diameter two

An induced forest k-partition of a graph G is a k  -partition (V1,V2,…,Vk)(V1,V2,…,Vk) of V(G)V(G) such that each G[Vi]G[Vi], 1⩽i⩽k1⩽i⩽k, 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.