  Three well-studied types of subgraph-restricted matchings are induced
matchings, uniquely restricted matchings, and acyclic matchings. While it is
hard to determine the maximum size of a matching of each of these types,
whether some given graph has a maximum matching that is induced or has a
maximum matching that is uniquely restricted, can both be decided efficiently.
In contrast to that we show that deciding whether a given bipartite graph of
maximum degree at most four has a maximum matching that is acyclic is
NP-complete. Furthermore, we show that maximum weight acyclic matchings can be
determined efficiently for $P_4$-free graphs and $2P_3$-free graphs, and we
characterize the graphs for which every maximum matching is acyclic.
