A graph G is {\em matching-decyclable} if it has a matching M such that
G−M is acyclic. Deciding whether G is matching-decyclable is an NP-complete
problem even if G is 2-connected, planar, and subcubic. In this work we
present results on matching-decyclability in the following classes: Hamiltonian
subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian
subcubic graphs we show that deciding matching-decyclability is NP-complete
even if there are exactly two vertices of degree two. For chordal and
distance-hereditary graphs, we present characterizations of
matching-decyclability that lead to O(n)-time recognition algorithms