Induced arboricity: Regular graphs and NP-completeness
Laurin Benz
Jonathan Rollin
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We prove that for each fixed k > 1 it is NP-hard to decide whether the edge set of a graph G can be covered by k subsets of edges such that each subset forms an induced forest in G. The problem stays NP-hard when restricted to bipartite input graphs and, in case 2 <= k <= 4, restricted to planar inputs. The smallest number k that makes such a cover possible is known as the induced arboricity of a graph. We also initiate studying the induced arboricity of regular graphs: For d-regular graphs we prove that their induced arboricity is at least d and, in case d >= 3, at most 2(d-1)2. The lower bound is tight and in case d <= 4 we improve the upper bound to d(d+1)/2 which is tight.

Jonathan Rollin | 10.05.2024