On grids in topological graphs

Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk
2014 Computational geometry  
A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straightline segment, then G is called a geometric graph. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that every n-vertex topological graph with no k-grid has O k (n) edges. We conjecture
more » ... hat the number of edges of every n-vertex topological graph with no k-grid such that all of its 2k edges have distinct endpoints is O k (n). This conjecture is shown to be true apart from an iterated logarithmic factor log * n. A k-grid is natural if its edges have distinct endpoints, and the arcs representing each of its edge subsets are pairwise disjoint. We also conjecture that every n-vertex geometric graph with no natural k-grid has O k (n) edges, but we can establish only an O k (n log 2 n) upper bound. We verify the above conjectures in several special cases.
doi:10.1016/j.comgeo.2014.02.003 fatcat:go2xryyqejcp7l4cjp6sdi54rq