  In this paper we investigate invertibility of graphs with a unique perfect
matching, i.e. graphs having a unique 1-factor. We recall the new notion of the
so-called negatively invertible graphs investigated by the authors in the
recent paper. It is an extension of the classical definition of an inverse
graph due to Godsil. We characterize all graphs with a unique perfect matching
on $m\le 6$ vertices with respect to their positive and negative invertibility.
We show that negatively invertible graphs exhibit properties like
selfinvertibility which cannot be observed within the class of positively
invertible non-bipartite graphs with a unique perfect matching.

Combinatorics