the node and link or chain objects of the graph occur or can be represented as though they occur upon a planar surface. Not more than one node may exist at any given point on the surface. Links or chains may only intersect at nodes.
A planar graph is a graph that can be drawn so that the edges only touch each other where they meet at vertices. You can usually re-draw a planar graph so that some of the edges cross. Even so, it is still a planar graph. When it is drawn so that the edges cross, the drawing is called a non-planar representation of a planar graph.
In graph theory, a planar graph is a graph that can be drawn (mathematicians say "can be embedded in the plane") so that no edges intersect. A nonplanar graph cannot be drawn without edge intersections.