网格域中弯曲距离的Reeb结构：带洞的循环界、圆盘上的精确扇区几何及双端口常数$c_2^\square=3$
The Reeb Structure of Bend Distance in Grid Domains: Cycle Bounds with Holes, Exact Sector Geometry on Disks, and the Two-Port Constant $c_2^\square=3$
AI总结：
本文研究带洞网格域中弯曲距离的Reeb结构，明确端点约定，推导带洞立方体域的循环界，证实无洞立方体圆盘的双端口树宽为3，为协同运动规划算法提供结构基础。
AI中文摘要：
针对离散简单多边形的协同运动规划的固定参数算法（ICALP 2026）基于单端口扇区分解：网格顶点被标记为到达定向终点所需的最小弯曲次数。我们在带洞的网格域上发展了该分解的结构理论。一旦明确其端点约定，弯曲距离就会扩展为域上的典范分段仿射函数，且扇区图是该函数的Reeb多重图的平行边投影。对于具有$h$个洞的有限纯平面立方体域，这给出$\beta_1(\Gamma_p) \le \beta_1(R_{f_p}) \le h$，以及可在线性时间内计算的最多包含$h$个扇区顶点的反馈集；坐标级的单洞示例表明，更精细的无洞几何（唯一前驱和直基线）不成立。在无洞立方体圆盘上，该机制是精确的：每个正扇区都是唯一直父界面的单侧拉伸，且双端口公共细化的树宽恰好为$c_2^\square=3$，尽管扇区数、循环秩和反馈数在此处已无界。端点精度是必要的而非装饰性的：根据原始定义的字面解读，一个无洞单位正方形的扇区图为$C_3$，这否定了单端口树引理；此处使用的增强约定正是原始分层过程计算的度量。我们未获得$f(k,h)n^{O(1)}$算法；本文为该程序提供了结构上的第一步和一个证伪工具。
英文摘要：
The fixed-parameter algorithm for coordinated motion planning on discretized simple polygons (ICALP 2026) rests on a single-port sector decomposition: grid vertices are labelled by the minimum number of bends needed to reach an oriented terminal. We develop the structure theory of this decomposition on grid domains with holes. Once its endpoint convention is made precise, bend distance extends to a canonical piecewise-affine function on the domain, and the sector graph is the parallel-edge shadow of the Reeb multigraph of that function. For a finite pure planar cubical domain with $h$ holes this yields $β_1(Γ_p) \le β_1(R_{f_p}) \le h$, together with a linear-time computable feedback set of at most $h$ sector vertices; coordinate-level one-hole examples show that the finer hole-free geometry -- unique predecessors and straight baselines -- fails. On hole-free cubical disks the machinery is exact: every positive sector is the one-sided extrusion of a unique straight parent interface, and the two-port common refinement has treewidth exactly $c_2^\square=3$, although sector count, cycle rank, and feedback number are unbounded already there. The endpoint precision is necessary rather than cosmetic: under the literal reading of the source's definition, one hole-free unit square has sector graph $C_3$, falsifying the single-port tree lemma; the augmented convention used here is exactly the metric computed by the source's own layering procedure. We do not obtain an $f(k,h)n^{O(1)}$ algorithm; the paper supplies the structural first step and a falsification tool for that program.
