On the vertex-arboricity of K5K5-minor-free graphs of diameter 2

An induced forest kk-partition of a graph GG is a kk-partition (V1,V2,…,Vk)(V1,V2,…,Vk) of the vertex set of GG such that, for each ii with 1≤i≤k1≤i≤k, the subgraph induced by ViVi is a forest. The vertex-arboricity of a graph GG is the minimum kk such that GG has an induced forest kk-partition. In the literature, it has been shown that every planar graph of diameter 2 has vertex-arboricity at most 2. The family of K5K5-minor-free graphs is a generalization of the planar graphs. We show in this paper that every K5K5-minor-free graph of diameter 2 has vertex-arboricity at most 2.