超越四度：近正交平面图绘制
Beyond Degree Four: Near-Orthogonal Planar Drawings
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中文总结 AI 辅助
研究最大度数无限制的图的平面折线图绘制，聚焦接近正交的绘制。证明测试至多\(h\)个非正交面的绘制问题是NP完全的，在固定和可变嵌入设置下，分别给出基于外平面性指数、\(h\)、树宽等参数的FPT算法及多项式时间近似方案。
中文摘要 AI 辅助
正交平面图绘制因其清晰性和广泛适用性，是图绘制中的经典主流研究课题。在图的正交平面图绘制中，每个面表示为正交多边形，即边为水平或垂直的多边形。然而，平面图当且仅当其最大度数至多为四时才允许这样的表示。本文考虑最大度数无限制的图的平面折线图绘制。聚焦于‘接近正交’的绘制，接近程度由非正交多边形的面的数量衡量。表明即使输入图是三连通的且有唯一平面嵌入，测试是否存在至多\(h\)个非正交面的平面折线图绘制问题是NP完全的。基于此计算难度，研究参数化和近似算法。在固定嵌入设置下，证明该问题对于（i）外平面性指数和（ii）自然参数\(h\)，允许线性时间FPT算法。此外，提供了一个由树宽参数化的FPT算法和一个多项式时间近似方案。在可变嵌入设置下，给出了一个由树宽参数化的双连通图的FPT算法。
英文摘要
Orthogonal planar drawings constitute a classical and mainstream research topic in graph drawing due to their clarity and wide applicability. In an orthogonal planar drawing of a graph, each face is represented as an orthogonal polygon, that is, a polygon whose edges are either horizontal or vertical. Yet a planar graph admits such a representation if and only if its maximum degree is at most four. In this paper, we consider planar polyline drawings of graphs with unrestricted maximum degree. We focus on drawings that are ``close to orthogonal'', where closeness is measured by the number of faces that are not orthogonal polygons. We show that, even when the input graph is triconnected and thus has a unique planar embedding, the problem of testing whether there exists a planar polyline drawing with at most $h$ non-orthogonal faces is NP-complete. Motivated by this computational hardness, we study parameterized and approximation algorithms. In the fixed-embedding setting, we prove that the problem admits linear-time FPT algorithms parameterized by (i) the outerplanarity index and (ii) the natural parameter $h$. In addition, we provide an FPT algorithm parameterized by the treewidth and a polynomial-time approximation scheme. In the variable-embedding setting, we give an FPT algorithm parameterized by treewidth for biconnected graphs.
