Embedding-Preserving Rectangle Visibility Representations of Nonplanar Graphs

A (weak) rectangle visibility representation, or simply an RVR, of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Given a graph with a fixed embedding in the plane, we show that the problem of testing whether this graph has an embedding-preserving RVR can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs, i.e., for embedded graphs having at most one crossing per edge. The linear time algorithm uses three forbidden configurations, which extend the set known for straight-line drawings of 1-plane graphs. The algorithm first checks for the presence of these forbidden configurations in the input graph, and then either an embedding-preserving RVR is computed (also in linear time) or a forbidden configuration is reported as a negative witness. Finally, we discuss extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.