Title: On some Graphs with a Unique Perfect Matching,
Abstract: We show that deciding whether a given graph $G$ of size $m$ has a unique
perfect matching as well as finding that matching, if it exists, can be done in
time $O(m)$ if $G$ is either a cograph, or a split graph, or an interval graph,
or claw-free. Furthermore, we provide a constructive characterization of the
claw-free graphs with a unique perfect matching.