Nonplanar Graph Drawings with k Vertices per Face

The study of nonplanar graph drawings with forbidden or desired crossing configurations has a long tradition in geometric graph theory, and received an increasing attention in the last two decades, under the name of beyond-planar graph drawing. In this context, we introduce a new hierarchy of graph families, called k(+)-real face graphs. For any integer k >= 1, a graph G is a k(+)-real face graph if it admits a drawing Gamma in the plane such that the boundary of each face (formed by vertices, crossings, and edges) contains at least k vertices of G. We give tight upper bounds on the maximum number of edges of k(+)- real face graphs. In particular, we show that 1(+)-real face and 2(+)-real face graphs with n vertices have at most 5n - 10 and 4n - 8 edges, respectively. Also, if all vertices are constrained to be on the boundary of the external face, then 1(+)-real face and 2(+)-real face graphs have at most 3n - 6 and 2.5n- 4 edges, respectively. We also study relationships between k(+)- real face graphs and beyond-planar graph families with hereditary property.