AI 中文总结
本文研究外k-平面图上图问题的参数化复杂性，证明Binary CSP等少数问题仍难解，多数问题固定参数可处理，还建立了外k-平面图的多项结构结果并明确其在图参数层次中的位置。
AI 中文摘要
若一个图存在直线画法，使得所有顶点都在一个圆上，且每条边最多被k条其他边交叉，则称该图为外k-平面图。我们以k为参数，研究外k-平面图上大量图问题的参数化复杂性。已知许多图问题以树宽或外平面性为参数时是XALP-难的，以路径宽为参数时是XNLP-难的。我们证明，仅Binary CSP（二元约束满足问题）和Scattered Set（分散集）等少数此类问题在外k-平面图上仍难解，而在给定输入图的外k-平面图画法的前提下，大量其他问题在此设定下成为固定参数可处理的，包括列表着色、容量支配集、容量顶点覆盖、目标出度定向、目标集选择等。除算法与复杂性结果外，我们还建立了若干结构结果：证明外k-平面图的mim-宽至多为k+2；证明割宽至多为k的图是外2k-平面图；证明反馈边集数至多为k的图是外6k-平面图。我们还证明许多图参数与外k-平面性不可比，从而明确了其在图参数层次结构中的位置。
英文摘要
A graph is outer k-planar if it admits a straight-line drawing in which all vertices lie on a circle and every edge is crossed by at most k other edges. We study the parameterized complexity of a broad collection of graph problems on outer k-planar graphs, with k as the parameter. Many graph problems are known to be XALP-hard when parameterized by treewidth or outerplanarity, and XNLP-hard when parameterized by pathwidth. We show that only a few such problems, including Binary CSP and Scattered Set, remain intractable on outer k-planar graphs, whereas a large class of the others become fixed-parameter tractable in this setting, assuming that an outer k-planar drawing of the input graph is given. These include List Coloring, Capacitated Dominating Set, Capacitated Vertex Cover, Target Outdegree Orientation, and Target Set Selection, among others. In addition to the algorithmic and complexity results, we establish several structural results. We show that outer k-planar graphs have mim-width at most k+2, that graphs of cut-width at most k are outer 2k-planar, and that graphs of feedback edge set number at most k are outer 6k-planar. We also show that many graph parameters are incomparable with outer k-planarity, thereby clarifying its position within the graph parameter hierarchy.
