Quasi-upward Planar Drawings with Minimum Curve Complexity

Graphs Drawn With Some Vertices per Face: Density and Relationships

“…Proof: Theorem 10 proves the existence of 1 + -real face graphs that are neither h-planar, nor min-h-planar, nor h-quasi planar. On the other hand, since the maximum number of edges of n-vertex h-planar graphs and min-h-planar graphs, for h ≥ 3, can be greater than 5n − 10 [20], [27], [28], [66], there exist h-planar graphs and min-h-planar graphs that are not 1 + -real face graphs (and hence that are not k + -real face graphs, for any k ≥ 2). Similarly, h-quasi planar graphs, for any h ≥ 3, can have higher density than 1 + -real face graphs, because 3-quasi planar graphs can have up to 6.5n − 20 edges [21].…”

“…For instance, for a given positive integer k, the family of k-planar graphs contains all graphs that admit a drawing with no more than k crossings per edge [19], [20], while k-quasi planar graphs are those that can be drawn without k mutually (i.e., pairwise) crossing edges [21]- [26]. A generalization of k-planar graphs, called min-k-planar graphs has been recently proposed, in which for any two crossing edges one of the two must contain at most k crossings [27], [28]. Other prominent examples of beyond-planar graph families are fan-planar graphs [29]- [35], where an edge is not allowed to cross two independent edges, and k-gap planar graphs (k ≥ 1) [36]- [38], where for each pair of crossing edges one of the two edges contains a small gap through which the other edge can pass, and only k gaps per edge are allowed.…”