I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree.

In particular I am most interested in results for planar graphs.

I have searched quite hard and there is a fairly extensive body of work on *vertex-arboricity* and *linear vertex arboricity* which relate to colouring a graph so that each colour class is a forest or a disjoint union of paths. (Rather than using the phrase "colouring the vertices", the vertex-arboricity etc is often defined as "partitioning the vertices" but of course they are the same thing.)

There are numerous extensions of the vertex-arboricity concept where the colour classes have to be nearly equal size, or when other restrictions are placed on the monochromatic subgraphs. In fact, put any of the words {equitable, list, star, critical, fractional} in front of "vertex-arboricity" and there is a paper on it.

The other relevant thing that came up was the phrase *Yutsis graphs* which refer to graphs that can be partitioned into two induced trees. However there are perhaps two or three papers on these, and they focussed primarily on cubic graphs, and did not help me.

So my question is:

*Is there an accepted graph-theoretic term and/or associated literature related to the concept of partitioning the vertices of a graph into induced trees?*

I am primarily concerned with exact structural results (things like characterisations of families of graphs that can be partitioned into very few trees) that I may be able to extend and/or adapt for my purposes.

I am not really interested in complexity results (e.g. NP-completeness) or anything to do with infinite graphs; there are numerous vertex-arboricity papers on complexity.