Topological Drawings of Complete Bipartite Graphs

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of complete bipartite graphs, in which we require that one side of the vertex set bipartition lies on the outer boundary of the drawing. We investigate the combinatorics of such drawings. For this purpose, we define combinatorial encodings of the drawings by enumerating the distinct drawings of subgraphs isomorphic to K2,2 and K3,2, and investigate the constraints they must satisfy. We prove in particular that for complete bipartite graphs of the form K2,n and K3,n, such an encoding corresponds to a drawing if and only if it obeys certain consistency conditions on triples and quadruples. For the case of K2,n, we make use of tools from the theory of pseudoline arrangements. For K3,n we show that drawings of the two induced K2,n can be manipulated to yield a drawing of K3,n. In the general case of Kk,n with k ≥ 2, we completely characterize and enumerate drawings in which the order of the edges around each vertex is the same for vertices on the same side of the bipartition. We also investigate straight-line drawings of K3,n and relate the realizability problem to that of stretchability of tropical pseudoline arrangements.