有界跨度的弱分层平面性
Weakly Leveled Planarity with Bounded Span
AI总结：
本文研究有界跨度的弱分层平面图绘制问题，证明其关于跨度参数 para-NP 难，并给出关于顶点覆盖数和树深的参数化复杂性结果，同时证明循环树等图类具有对数级或常数级跨度上界。
摘要：
本文研究图的平面绘制，其中每个顶点表示为沿一系列水平线（称为层）上的一个点，每条边要么是水平线段，要么是严格 $y$-单调曲线。如果一个图允许这样的绘制且边的跨度至多为 $s$，则称其为 $s$-跨度弱分层平面图；一条边的跨度是它所触及的层数减一。我们从计算和组合两个角度研究计算 $s$-跨度弱分层平面绘制的问题。我们证明该问题关于其自然参数 $s$ 是 para-NP 难的，并研究其关于广泛使用的结构参数的复杂性。我们证明了关于顶点覆盖数存在多项式大小的核，并证明当以树深为参数时该问题是 FPT 的。我们还给出了各种图类的跨度的上界和下界。值得注意的是，我们证明循环树（cycle trees）——一类推广 Halin 图的 $2$-外平面图族——是 $Θ(\log n)$-跨度弱分层平面的，并且在 $3$-连通时是 $4$-跨度弱分层平面的。作为这些组合结果的副产品，我们得到了所考虑的图族的边长比的改进界。
英文摘要：
This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly $y$-monotone curve. A graph is $s$-span weakly leveled planar if it admits such a drawing where the edges have span at most $s$; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing $s$-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter $s$ and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of $2$-outerplanar graphs generalizing Halin graphs, are $Θ(\log n)$-span weakly leveled planar and $4$-span weakly leveled planar when $3$-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.
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