On grids in topological graphs

Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk
2009 Proceedings of the 25th annual symposium on Computational geometry - SCG '09  
A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. 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 for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this remains true even when: (1) considering grids with distinct vertices; or (2) all
more » ... ertices; or (2) all edges are straight-line segments and the edges within each subset of the grid are required to be pairwise disjoint. These conjectures are shown to be true apart from log * n and log 2 n factors, respectively. We also settle the conjectures for some special cases.
doi:10.1145/1542362.1542430 dblp:conf/compgeom/AckermanFPS09 fatcat:lt6ikax4krb5laweowxm7ep4q4