  In 2005, Goddard, Hedetniemi, Hedetniemi and Laskar [Generalized
subgraph-restricted matchings in graphs, Discrete Mathematics, 293 (2005) 129 -
138] asked the computational complexity of determining the maximum cardinality
of a matching whose vertex set induces a disconnected graph. In this paper we
answer this question. In fact, we consider the generalized problem of finding
$c$-disconnected matchings; such matchings are ones whose vertex sets induce
subgraphs with at least $c$ connected components. We show that, for every fixed
$c \geq 2$, this problem is NP-complete even if we restrict the input to
bounded diameter bipartite graphs, while can be solved in polynomial time if $c
= 1$. For the case when $c$ is part of the input, we show that the problem is
NP-complete for chordal graphs, while being solvable in polynomial time for
interval graphs. Finally, we explore the parameterized complexity of the
problem. We present an FPT algorithm under the treewidth parameterization, and
an XP algorithm for graphs with a polynomial number of minimal separators when
parameterized by $c$. We complement these results by showing that, unless NP
$\subseteq$ coNP/poly, the related Induced Matching problem does not admit a
polynomial kernel when parameterized by vertex cover and size of the matching
nor when parameterized by vertex deletion distance to clique and size of the
matching. As for Connected Matching, we show how to obtain a maximum connected
matching in linear time given an arbitrary maximum matching in the input.
