  This paper introduces a notion of presentation for locally inverse semigroups
and develops a graph structure to describe the elements of locally inverse
semigroups given by these presentations. These graphs will have a role similar
to the role that Cayley graphs have for group presentations or that
Sch\"utzenberger graphs have for inverse monoid presentations. However, our
graphs have considerable differences with the latter two, even though locally
inverse semigroups generalize both groups and inverse semigroups. For example,
the graphs introduced here are not `inverse word graphs'. Instead, they are
bipartite graphs with both oriented and non-oriented edges, and with labels on
the oriented edges only. A byproduct of the theory developed here is the
introduction of a graphical method for dealing with general locally inverse
semigroups. These graphs are able to describe, for a locally inverse semigroup
given by a presentation, many of the usual concepts used to study the structure
of semigroups, such as the idempotents, the inverses of an element, the Green's
relations, and the natural partial order. Finally, the paper ends
characterizing the semigroups belonging to some usual subclasses of locally
inverse semigroups in terms of properties on these graphs.
