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On the hardness of deciding the equality of the induced and the uniquely restricted matching number
If denotes the subgraph of a graph induced by the set of vertices
that are covered by some matching in , then is an induced or a
uniquely restricted matching if is -regular or if is the unique
perfect matching of , respectively. Let and
denote the maximum cardinality of an induced and a uniquely restricted matching
in . Golumbic, Hirst, and Lewenstein (Uniquely restricted matchings,
Algorithmica 31 (2001) 139-154) posed the problem to characterize the graphs
with . We prove that the corresponding decision
problem is NP-hard, which suggests that a good characterization is unlikely to
be possible