Graphs Drawn With Some Vertices per Face: Density and Relationships

• Published in: IEEE access Vol. 12; pp. 68828 - 68846
• Main Authors: Binucci, Carla, Di Battista, Giuseppe, Didimo, Walter, Dujmovic, Vida, Hong, Seok-Hee, Kaufmann, Michael, Liotta, Giuseppe, Morin, Pat, Tappini, Alessandra
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Apexes; Beyond-planar graph drawing; Computational modeling; Density; edge density; Face recognition; geometric graph theory; Graph drawing; Graph theory; graph visualization; Graphical representations; Graphs; inclusion relationships; Surveys; Upper bound; Upper bounds; Visualization
• DOI: 10.1109/ACCESS.2024.3401078
• ISSN: 2169-3536

Graph drawing beyond planarity is a research area that has received an increasing attention in the last twenty years, driven by the necessity to mitigate the visual complexity inherent in geometric representations of non-planar graphs. This research area stems from the study of graph layouts with forbidden crossing configurations, a well-established subject in geometric and topological graph theory. In this context, the contribution of this paper is as follows: 1) We introduce a new hierarchy of graph families, called <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face graphs; for any integer <inline-formula> <tex-math notation="LaTeX">k \geq 1 </tex-math></inline-formula>, a graph G is a <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face graph if it admits a drawing <inline-formula> <tex-math notation="LaTeX">\Gamma </tex-math></inline-formula> in the plane such that the boundary of each face (formed by vertices, crossings, and edges) contains at least k vertices of G ("<inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>" stands for k or more); 2) We give tight upper bounds on the edge density of <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face graphs, namely we prove that n-vertex <inline-formula> <tex-math notation="LaTeX">1^{+} </tex-math></inline-formula>-real face and <inline-formula> <tex-math notation="LaTeX">2^{+} </tex-math></inline-formula>-real face graphs have at most <inline-formula> <tex-math notation="LaTeX">5n-10 </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">4n-8 </tex-math></inline-formula> edges, respectively. Furthermore, in a constrained scenario in which all vertices must lie on the boundary of the external face, <inline-formula> <tex-math notation="LaTeX">1^{+} </tex-math></inline-formula>-real face and <inline-formula> <tex-math notation="LaTeX">2^{+} </tex-math></inline-formula>-real face graphs have at most <inline-formula> <tex-math notation="LaTeX">3n-6 </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">2.5n-4 </tex-math></inline-formula> edges, respectively; 3) We characterize the complete graphs that admit a <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face drawing or an outer <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face drawing for any <inline-formula> <tex-math notation="LaTeX">k \geq 1 </tex-math></inline-formula>. We also provide a clear picture for the majority of complete bipartite graphs; and 4) We establish relationships between <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face graphs and other prominent beyond-planar graph families; notably, we show that for any <inline-formula> <tex-math notation="LaTeX">k \geq 1 </tex-math></inline-formula>, the class of <inline-formula> <tex-math notation="LaTeX">k^{+} </tex-math></inline-formula>-real face graphs is not included in any family of beyond-planar graphs with hereditary property.