On 3D visibility representations of graphs with few crossings per edge

A graph is k-planar if it can be drawn in the plane such that each edge is crossed at most k>0 times. These graphs represent a natural extension of planar graphs and they are among the most investigated families in the growing field of graph drawing beyond planarity. In this paper, we study visibility representations of k-planar graphs in three dimensions. In particular, we provide a technique for a meaningful family of 2-planar graphs, called 5-kite-augmented graphs, which include as subgraphs all 1-planar graphs and all simple optimal 2-planar graphs (i.e., those 2-planar graphs that attain the maximum number of edges). We prove that every 5-kite-augmented graph has a z-parallel visibility representation, i.e., a three-dimensional visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of the input graph, which is a well-known type of visibility representation in two dimensions.