We study the following question, which has been considered since the 90's: Does every $st$-planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed $y$-coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of $st$-planar graphs that always admit a planar straight-line dominance drawing. These include $st$-planar $3$-trees in which every stacking operation introduces two edges incoming into the new vertex, $st$-planar graphs in which every vertex is adjacent to the sink, $st$-planar graphs in which no face has the left boundary that is a single edge, and $st$-planar graphs that have a leveling with span at most two.

翻译：我们研究以下自20世纪90年代以来一直被探讨的问题：是否每个$st$-平面图都允许存在平面直线支配图？我们通过证明以下结论，为该问题的难度提供了具体证据：与上向平面直线图不同，具有指定$y$坐标的平面直线支配图并不总是存在，且平面直线支配图无法总是通过收缩-绘制-扩展的归纳方法构建。我们还展示了几类总是允许存在平面直线支配图的$st$-平面图。这些包括：每个堆叠操作引入两条指向新顶点的边的$st$-平面$3$-树、每个顶点都与汇点相邻的$st$-平面图、没有面其左边界为单一边的$st$-平面图，以及具有跨度至多为二的层级划分的$st$-平面图。