Simultaneous visibility representations of plane st-graphs using L-shapes

Let 〈Gr,Gb〉 be a pair of plane st-graphs with the same vertex set V. A simultaneous visibility representation with L-shapes of 〈Gr,Gb〉 is a pair of bar visibility representations 〈Γr,Γb〉 such that, for every vertex v∈V, Γr(v) and Γb(v) are a horizontal and a vertical segment, respectively, which share an end-point. In other words, every vertex is drawn as an L-shape, every edge of Gr is a vertical visibility segment, and every edge of Gb is a horizontal visibility segment. Also, no two L-shapes intersect each other. An L-shape has four possible rotations, and we assume that each vertex is given a rotation for its L-shape as part of the input. Our main results are: (i) a characterization of those pairs of plane st-graphs admitting such a representation, (ii) a quadratic time algorithm to recognize them, and (iii) a linear time drawing algorithm if the test is positive. As an application, starting from a simultaneous visibility representation with L-shapes, we show how to compute a simultaneous embedding of the two graphs with at most two bends per edge and right-angle crossings.