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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≥2\documentclass[12pt]{minimal}
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\begin{document}$$c \ge 2$$\end{document}, this problem is NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {NP}$$\end{document}-complete\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {complete}$$\end{document} even if we restrict the input to bounded diameter bipartite graphs. For the case when c is part of the input, we show that the problem is NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {NP}$$\end{document}-complete\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {complete}$$\end{document} for chordal graphs while being solvable in polynomial time for interval graphs, FPT\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {FPT}$$\end{document} when parameterized by treewidth, and XP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {XP}$$\end{document} for graphs with a polynomial number of minimal separators, when parameterized by c.