We consider the problem of determining if a pair of undirected graphs $〈G_{\mathsf{v}},G_{\mathsf{h}}〉$, 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 $G_{\mathsf{v}}$ (respectively, $G_{\mathsf{h}}$). 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 $G_{\mathsf{v}}$ (respectively, $G_{\mathsf{h}}$). 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.

我们考虑确定一对无向图的问题 $〈G_{\mathsf{v}}，G_{\mathsf{H}}〉$共享相同的顶点集，并使用不透明的几何形状表示顶点，并使用形状之间的垂直（分别是水平）可见性来确定边的边缘 $G_{\mathsf{v}}$ （分别， $G_{\mathsf{H}}$）。如果提供了每个边缘所需的可见性的方向（并且顶点形状足够简单），则可以有效地确定两个图形的这种同时可见性表示形式，但不清楚边缘方向是否对效率至关重要。在此，从u指向v的边表示代表u的形状在v in的形状下方（分别在其左侧）$G_{\mathsf{v}}$ （分别， $G_{\mathsf{H}}$）。我们证明了，即使没有比路径稍微复杂的图形，问题也是没有信息的NP完全问题。此外，我们使用固定方向的L形来表征哪些路径对具有同时可见性表示。这缩小了可能的图形族的范围，对于这些图形族来说，确定同时可见性表示并非无关紧要，但不是NP困难的。