Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

On fan-crossing and fan-crossing free graphs

“…The node-to-circle expansion operation preserves more properties, such as 3connectivity and the genus, and it may simplify graphs. For example, complete bipartite graphs K 4,n are fan-crossing but are not k-planar for k = 0, 1, 2, 3, 4 and n = 3, 5, 7, 10, 19, respectively, as shown by Angelini et al [9]. Also K 4,7 is not fan-crossing free.…”

“…For fan-planar graphs, the following has been proved [9,14,16,19,42]. The shown results also hold for fan-crossing graphs, since the restriction from configuration II is not used in the proofs.…”

“…They focus on the density of (k)-fan-crossing free graphs. Complete and complete bipartite graphs were studied by Angelini et al [9]. The state of the art is as follows, see also [31].…”

On Optimal Beyond-Planar Graphs

“…Beyond-planar graphs have been studied with different intensity and depth. In particular, the density, which is an upper bound on the number of edges of n-vertex graphs, the size of the largest complete (bipartite) graph [8], and inclusion relations have been investigated [29,36]. The linear density is a typical property of beyond-planar graphs, see Table 1.…”

“…The linear density is a typical property of beyond-planar graphs, see Table 1. Small complete graphs K k with k ≤ 11 distinguish some types, see [8,21]. Inclusions are canonical, in general, so that a restriction on drawings implies a proper inclusion for the graph classes [29].…”