The Reeb Structure of Bend Distance in Grid Domains: Cycle Bounds with Holes, Exact Sector Geometry on Disks, and the Two-Port Constant $c_2^\square=3$

Abstract

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 $\beta_1(\Gamma_p) \le \beta_1(R_{f_p}) \le h$, 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.