Windrose Planarity

Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Valentino Di Donato, Philipp Kindermann, Günter Rote, Ignaz Rutter
2018 ACM Transactions on Algorithms  
Given a planar graph G and a partition of the neighbors of each vertex v in four sets UR(v), UL(v), DL(v), and DR(v), the problem Windrose Planarity asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ UR(v) is above and to the right of v, (ii) each neighbor u ∈ UL(v) is above and to the left of v, (iii) each neighbor u ∈ DL(v) is below and to the left of v, (iv) each neighbor u ∈ DR(v) is below and to the right of v, and (v) edges
more » ... re represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is NP-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a given combinatorial embedding. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph with n vertices that has a windrose-planar drawing, we can construct one with at most one bend per edge and with at most 2n-5 bends in total, which lies on the 3n × 3n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.
doi:10.1145/3239561 fatcat:m4ys24l7szcfha5ckrbencenca