Simultaneous visibility representations of undirected pairs of graphs

We consider the problem of determining if a pair of undirected graphs (Gv, Gh), which share the same vertex set, has a representation using opaque geometric shapes for vertices, and vertical (respectively, horizontal) visibility between shapes to determine the edges of Gv (respectively, Gh). While such a simultaneous visibility representation of two graphs can be determined efficiently if the direction of the required visibility for each edge is provided (and the vertex shapes are sufficiently simple), it was unclear if edge direction is critical for efficiency. Here, an edge directed from u to v implies that the shape representing u is below (respectively, left of) the shape for v in Gv (respectively, Gh). We show that the problem is NP-complete without that information, even for graphs that are only slightly more complex than paths. In addition, we characterize which pairs of paths have simultaneous visibility representations using fixed orientation L-shapes. This narrows the range of possible graph families for which determining simultaneous visibility representation is non-trivial yet not NP-hard. (C) 2021 Elsevier B.V. All rights reserved.