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

An induced forest k -partition of a graph G is a k -partition ( V 1 , V 2 , … , V k ) of the vertex set of G such that, for each i with 1 ≤ i ≤ k , the subgraph induced by V i is a forest. The vertex-arboricity of a graph G is the minimum k such that G has an induced forest k -partition. In the literature, it has been shown that every planar graph of diameter 2 has vertex-arboricity at most 2. The family of K 5 -minor-free graphs is a generalization of the planar graphs. We show in this paper that every K 5 -minor-free graph of diameter 2 has vertex-arboricity at most 2.