  A generic immersion of a planar graph into the 2-space is said to be knotted
if there does not exist a trivial embedding of the graph into the 3-space
obtained by lifting the immersion with respect to the natural projection from
the 3-space to the 2-space. In this paper we show that if a generic immersion
of a planar graph is knotted then the number of double points of the immersion
is more than or equal to three. To prove this, we also show that an embedding
of a graph obtained from a generic immersion of the graph (does not need to be
planar) with at most three double points is totally free if it contains neither
a Hopf link nor a trefoil knot.
