🤖 AI Summary
This study investigates the conditions under which drawing all edges of a complete graph in the plane with curves necessarily yields a large set of pairwise non-crossing curves—that is, a planar substructure—under specific crossing rules. By integrating combinatorial geometry and graph-theoretic techniques with crossing analysis and constructive proofs, the work provides the first systematic characterization of two mild crossing constraints that inevitably force the emergence of numerous disjoint curves. Moreover, it constructs an explicit drawing scheme in which every pair of curves intersects, while simultaneously satisfying tight upper and lower bounds on the total number of crossings. These results uncover a profound connection between local crossing rules and global planar structure, offering new structural insights and constructive tools for graph drawing theory.
📝 Abstract
This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is obeyed: any two adjacent curves do not cross, or any two non-adjacent curves cross at most once. Here, two curves are called adjacent if they share an endpoint. On the other hand, we demonstrate how to draw all curves such that any two adjacent curves cross exactly once, any two non-adjacent curves cross at least once and at most twice, and thus no two curves are disjoint. Furthermore, we analyze the emergence of disjoint curves without these mild assumptions, and characterize the plane structures in complete graph drawings guaranteed by each of the rules above.