  Extending the work of Godsil and others, we investigate the notion of the
inverse of a graph (specifically, of bipartite graphs with a unique perfect
matching). We provide a concise necessary and sufficient condition for the
invertibility of such graphs and generalize the notion of invertibility to
multigraphs. We examine the question of whether there exists a "litmus
subgraph" whose bipartiteness determines invertibility. As an application of
our invertibility criteria, we quickly describe all invertible unicyclic
graphs. Finally, we describe a general combinatorial procedure for iteratively
constructing invertible graphs, giving rise to large new families of such
graphs.
