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

TL;DR

This paper characterizes, recognizes, and efficiently constructs simultaneous visibility representations of plane st-graph pairs using L-shapes, enabling clear visualization of complex graph relationships.

Contribution

It provides a characterization, recognition algorithm, and linear-time drawing method for L-shape based simultaneous visibility representations of plane st-graph pairs.

Findings

Characterization of graph pairs admitting L-shape visibility representations

Cubic time recognition algorithm for such pairs

Linear time construction algorithm for valid representations

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 $\langle\Gamma_r,\Gamma_b\rangle$ such that, for every vertex $v \in V$, $\Gamma_r(v)$ and $\Gamma_b(v)$ 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.