On Drawings and Decompositions of 1-Planar Graphs

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. We show that every 1-planar drawing of any 1-planar graph on n vertices has at most n − 2 crossings; moreover, this bound is tight. By this novel necessary condition for 1-planarity, we characterize the 1-planarity of Cartesian product Km × Pn. Based on this condition, we also derive an upper bound on the number of edges of bipartite 1-planar graphs, and we show that each subgraph of an optimal 1-planar graph (i.e., a 1-planar graph with n vertices and 4n− 8 edges) can be decomposed into a planar graph and a forest.