Induced and weak induced arboricities

Abstract

We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G.For a class F of graphs and a graph parameter p, let p(F)=sup{p(G)∣G∈F}. We show that ia(F) is bounded from above by an absolute constant depending only on F, that is ia(F)≠∞ if and only if χ(F∇12)≠∞, where F∇12 is the class of 12-shallow minors of graphs from F andχ is the chromatic number.As a main contribution of this paper, we provide bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8≤ia(F)≤10.In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.