  We refine and advance the study of the local structure of idempotent finite
algebras started in [A.Bulatov, The Graph of a Relational Structure and
Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like
structure on an arbitrary finite idempotent algebra including those admitting
type 1. We show that this graph is connected, its edges can be classified into
4 types corresponding to the local behavior (set, semilattice, majority, or
affine) of certain term operations. We also show that if the variety generated
by the algebra omits type 1, then the structure of the algebra can be
`improved' without introducing type 1 by choosing an appropriate reduct of the
original algebra. Taylor minimal idempotent algebras introduced recently is a
special case of such reducts. Then we refine this structure demonstrating that
the edges of the graph of an algebra omitting type 1 can be made `thin', that
is, there are term operations that behave very similar to semilattice,
majority, or affine operations on 2-element subsets of the algebra. Finally, we
prove certain connectivity properties of the refined structures.
  This research is motivated by the study of the Constraint Satisfaction
Problem, although the problem itself does not really show up in this paper.
