  We study the graphs formed from instances of the stable matching problem by
connecting pairs of elements with an edge when there exists a stable matching
in which they are matched. Our results include the NP-completeness of
recognizing these graphs, an exact recognition algorithm that is singly
exponential in the number of edges of the given graph, and an algorithm whose
time is linear in the number of vertices of the graph but exponential in a
polynomial of its carving width. We also provide characterizations of graphs of
stably matchable pairs that belong to certain classes of graphs, and of the
lattices of stable matchings that can have graphs in these classes.
