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A generic immersion of a planar graph into the 2-space is said to be knottedif there does not exist a trivial embedding of the graph into the 3-spaceobtained by lifting the immersion with respect to the natural projection fromthe 3-space to the 2-space. In this paper we show that if a generic immersionof a planar graph is knotted then the number of double points of the immersionis more than or equal to three. To prove this, we also show that an embeddingof a graph obtained from a generic immersion of the graph (does not need to beplanar) with at most three double points is totally free if it contains neithera Hopf link nor a trefoil knot.

Regular projections of graphs with at most three double points