Decycling with a matching

As a natural variant of the many decycling notions studied in graphs, we consider the problem to decide whether a given graph G has a matching M such that G−M is a forest. We establish NP-completeness of this problem for 2-connected planar subcubic graphs, and describe polynomial time algorithms that also determine such a matching if it exists for graphs that are claw- and paw-free, P5-free, chordal, and C4-free distance hereditary.