  We give a characterization of the effect of sequences of pivot operations on
a graph by relating it to determinants of adjacency matrices. This allows us to
deduce that two sequences of pivot operations are equivalent iff they contain
the same set S of vertices (modulo two). Moreover, given a set of vertices S,
we characterize whether or not such a sequence using precisely the vertices of
S exists. We also relate pivots to perfect matchings to obtain a
graph-theoretical characterization. Finally, we consider graphs with self-loops
to carry over the results to sequences containing both pivots and local
complementation operations.
