  We characterize the graphs $G$ for which their toric ideals $I_G$ are
complete intersections. In particular we prove that for a connected graph $G$
such that $I_G$ is complete intersection all of its blocks are bipartite except
of at most two. We prove that toric ideals of graphs which are complete
intersections are circuit ideals. The generators of the toric ideal correspond
to even cycles of $G$ except of at most one generator, which corresponds to two
edge disjoint odd cycles joint at a vertex or with a path. We prove that the
blocks of the graph satisfy the odd cycle condition. Finally we characterize
all complete intersection toric ideals of graphs which are normal.
