Simultaneous Visibility Representations of Plane st-graphs Using L-shapes

Abstract

Let $$\langle G_r,G_b \rangle $$ be a pair of plane st-graphs with the same vertex set V. A simultaneous visibility representation with L-shapes of $$\langle G_r,G_b \rangle $$ is a pair of bar visibility representations $$\langle \varGamma _r,\varGamma _b\rangle $$ such that, for every vertex $$v \in V$$, $$\varGamma _rv$$ and $$\varGamma _bv$$ are a horizontal and a vertical segment, which share an end-point. In other words, every vertex is drawn as an L-shape, every edge of $$G_r$$ is a vertical visibility segment, and every edge of $$G_b$$ 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 cubic time algorithm to recognize them, and iii a linear time drawing algorithm if the test is positive.