We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $\mathbb{R}^3$. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $K_5$, $K_{5,81}$, or any nonplanar $3$-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $K_{4,4}$, and $K_{3,5}$ can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable $n$-vertex graphs is in $\Omega(n \log n)$. From the non-realizability of $K_{5,81}$, we obtain that any realizable $n$-vertex graph has $O(n^{9/5})$ edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

翻译：我们研究给定图是否可以在$\mathbb{R}^3$中实现为多面体表面多边形单元的邻接图。我们证明每个图都可以实现为具有任意多边形单元的多面体表面，但如果要求单元是凸的，则并非如此。特别地，如果给定图包含$K_5$、$K_{5,81}$或任何非平面$3$-树作为子图，则不存在这样的实现。另一方面，所有平面图、$K_{4,4}$和$K_{3,5}$都可以用凸单元实现。对于每条边至少细分一次的任意图的任何细分，以及根据McMullen等人（1983）的结果，对于任何超立方体，同样成立。我们的结果对描述具有凸单元的多面体表面的图的最大密度有影响：超立方体的可实现性表明，所有可实现的$n$顶点图的最大边数在$\Omega(n \log n)$中。从$K_{5,81}$的不可实现性，我们得到任何可实现的$n$顶点图最多有$O(n^{9/5})$条边。因此，这些图可以比平面图密集得多，但不能任意密集。