The k-strong induced arboricity of a graph

The induced arboricity of a graph G is the smallest number of induced forests covering the edges of G. This is a well-defined parameter bounded from above by the number of edges of G when each forest in a cover consists of exactly one edge. Not all edges of a graph necessarily belong to induced forests with larger components. For k > 1, we call an edge k-valid if it is contained in an induced tree on k edges. The k-strong induced arboricity of G, denoted by fk(G), is the smallest number of induced forests with components of sizes at least k that cover all k-valid edges in G. This parameter is highly non-monotone. However, we prove that for any proper minorclosed graph class C, and more generally for any class of bounded expansion, and any k > 1, the maximum value of fk(G) for G ∈ C is bounded from above by a constant depending only on C and k. We prove that f2(G) 6 3 ( t+1 3 ) for any graph G of tree-width t and that fk(G) 6 (2k) for any graph of tree-depth d. In addition, we prove that f2(G) 6 310 when G is planar, which implies that the maximum adjacent closed vertex-distinguishing chromatic number of planar graphs is constant.