The Hanani-Tutte Theorem and Non-separating Planar Graphs

Graphs are a useful data structure for representing networks of relationships among objects. Drawings of graphs help humans to comprehend graphs better. We study several types of drawings such as planar drawings, in which edges do not cross, and superthrackles, in which any two edges cross. We show how to reduce the number of crossings in certain drawings to obtain a planar drawing or a superthrackle drawing. Moreover, we introduce a class of planar graphs that cannot be disconnected by removal of a cycle, determine its fundamental properties and use it to prove results about crossings in drawings of graphs.