What is the reason to study planar graphs and algorithms on such graphs (as well as algorithms allowing to check a graph's planarity)? Where in industry this knowlege is needed? I know that planarity arises in microchip design ("wires" intersection). I also found the following excerpt from a lecture notes:
The study of two-dimensional images often results in problems related to planar graphs, as does the solution of many problems on the two-dimensional surface of our earth. Many natural three-dimensional graphs arise in scientific and engineering problems. These often come from well-shaped meshes, which share many properties with planar graphs.
Where I can read about it in more detail? Or may be you can explain in a few words what exactly these applications are?