无向图对的同时可见性表示
Simultaneous Visibility Representations of Undirected Pairs of Graphs
摘要：
我们考虑的问题，确定一对无向图$\langle G_\mathsf{V}, G_\mathsf{H} \rangle$，其中共享相同的顶点集，具有使用不透明的几何形状的顶点，和形状之间的垂直/水平的可见性，以确定边缘的$G_\mathsf{V}$ / $G_\mathsf{H}$的表示。虽然如果提供了每个边的所需可见性的方向（并且顶点形状足够简单），则可以有效地确定两个图的这种同时可见性表示，但是不清楚边方向对于效率是否是关键的。我们表明，这个问题是$\mathsf{NP}$ -完全没有这些信息，即使是图，只是稍微复杂的路径。此外，我们的特点，这对路径具有同时使用固定方向L-形状的可见性表示。这缩小了可能的图形族的范围，对于这些图形族，确定同时可见性表示是不平凡的，但不是$\mathsf{NP}$困难的。
英文摘要：
We consider the problem of determining if a pair of undirected graphs $\langle G_\mathsf{V}, G_\mathsf{H} \rangle$, which share the same vertex set, has a representation using opaque geometric shapes for vertices, and vertical/horizontal visibility between shapes to determine the edges of $G_\mathsf{V}$/$G_\mathsf{H}$. 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. We show that the problem is $\mathsf{NP}$-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 $\mathsf{NP}$-hard.
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