A finite set $P$ of points in the plane is $n$-universal with respect to a class $\mathcal{C}$ of planar graphs if every $n$-vertex graph in $\mathcal{C}$ admits a crossing-free straight-line drawing with vertices at points of $P$. For the class of all planar graphs the best known upper bound on the size of a universal point set is quadratic and the best known lower bound is linear in $n$. Some classes of planar graphs are known to admit universal point sets of near linear size, however, there are no truly linear bounds for interesting classes beyond outerplanar graphs. In this paper, we show that there is a universal point set of size $2n-2$ for the class of bipartite planar graphs with $n$ vertices. The same point set is also universal for the class of $n$-vertex planar graphs of maximum degree $3$. The point set used for the results is what we call an exploding double chain, and we prove that this point set allows planar straight-line embeddings of many more planar graphs, namely of all subgraphs of planar graphs admitting a one-sided Hamiltonian cycle. The result for bipartite graphs also implies that every $n$-vertex plane graph has a $1$-bend drawing all whose bends and vertices are contained in a specific point set of size $4n-6$, this improves a bound of $6n-10$ for the same problem by L\"offler and T\'oth.

翻译：有限平面点集 $P$ 关于平面图类 $\mathcal{C}$ 是 $n$-通用的，如果 $\mathcal{C}$ 中每个 $n$ 顶点图均存在顶点位于 $P$ 中点的无交叉直线绘制。对于所有平面图构成的类，已知通用点集大小的最优上界是二次的，最优下界则是 $n$ 的线性函数。已知某些平面图类具有近线性规模的通用点集，但对于外平面图之外的有趣图类，尚未获得真正线性的界限。本文证明，对具有 $n$ 个顶点的二部平面图类，存在大小为 $2n-2$ 的通用点集。同一组点集对最大度为 $3$ 的 $n$ 顶点平面图类也是通用的。本文使用的点集被称为“爆炸双链”，并证明该点集允许更多平面图（即所有具有单侧哈密顿圈的平面图的子图）进行平面直线嵌入。对二部图的结果还蕴含：每个 $n$ 顶点平面图存在一种 $1$ 折绘制，其所有折点和顶点均包含于大小为 $4n-6$ 的特定点集中，这改进了 L\"offler 和 T\'oth 关于同一问题的 $6n-10$ 界。