Induced and Weak Induced Arboricities
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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. This notion generalizes the classical notions of the arboricity and strong chromatic index. For a class F of graphs and a graph parameter p, let p(F) = {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∇1/2) ≠∞, where F∇1/2 is the class of 1/2-shallow minors of graphs from F and χ is the chromatic number. Further, we give 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.
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