On the Perspectives Opened by Right Angle Crossing Drawings

Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of a drawing if edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straight-line upward RAC drawings require exponential area. Exploiting the techniques introduced for studying straight-line upward RAC drawings, we also show that there exist planar undirected graphs requiring quadratic area in any straight-line RAC drawing. Second, we study whether RAC drawings allow us to draw boundeddegree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertexdegree at most six (at most three) admits a RAC drawing with curve complexity two (resp. one) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings.