Plane Covering Paths in Computational Geometry

• Plane covering paths are polygonal paths defined to cover every point in a plane, often allowing Steiner points and non-crossing constraints to minimize segments.
• The concept varies by context: computational geometry focuses on segment minimization, reconfiguration theory studies non-crossing Hamiltonian paths, and robotics targets obstacle-aware coverage.
• Recent advances include improved extremal bounds and O(n log n) algorithms employing techniques like cap/cup extraction and strip decomposition for optimal segment reductions.

A plane covering path is a polygonal path in the plane whose straight-line segments collectively cover a prescribed geometric input, but the term is not uniform across the literature. In one standard computational-geometric sense, a covering path for a finite point set $P$ is a polygonal path such that every point of $P$ lies on a segment of the path; the path vertices need not belong to $P$, so Steiner points are allowed, and the plane or non-crossing variant forbids segment crossings in the interior (Dumitrescu et al., 2013, Akitaya et al., 9 Jul 2025). In a second sense, used in reconfiguration theory, “plane covering path” is synonymous with a plane straight-line spanning path or non-crossing Hamiltonian path whose vertex set is exactly the given planar point set (Aichholzer et al., 2022). A third usage appears in robotics, where coverage path planning seeks near-complete coverage of a planar domain, often with obstacles, rather than coverage of a finite point set by a polygonal chain (Lin et al., 2017).

1. Definitions and terminological scope

For a finite point set $P\subset \mathbb{R}^2$, a covering path is a straight-line polygonal path such that every point of $P$ lies either at a vertex of the path or on one of its edges; the vertices of the path are not required to be points of $P$ (Biniaz, 2023). The path is plane or non-crossing if its segments do not cross each other; segments may meet at common endpoints (Akitaya et al., 9 Jul 2025). In this model, the optimization variable is the number of straight segments, often called links or edges, not Euclidean length (Dumitrescu et al., 2013).

This notion differs from a spanning path. Every spanning path is a covering path, but a covering path may have fewer than $n-1$ edges because points can be covered by lying anywhere on an edge, not necessarily at a vertex (Biniaz, 2023). In general position, where no three points are collinear, each segment can cover at most two points, which yields the trivial lower bound $\lceil n/2\rceil$, while a non-crossing spanning path gives the trivial upper bound $n-1$ (Biniaz, 2023).

A standard worst-case extremal function is

$\pi(n),$

defined as the minimum integer $P$0 such that every set of $P$1 points in the plane can be covered by a non-crossing path with at most $P$2 edges (Biniaz, 2023). The analogous function for non-crossing covering trees is denoted by $P$3 (Biniaz, 2023).

The Hamiltonian-path usage is formally different. For a planar point set $P$4 in general position, $P$5 denotes the set of all plane straight-line spanning paths with vertex set $P$6; here every point of $P$7 is a vertex, and the edges are pairwise non-crossing (Aichholzer et al., 2022). The source explicitly notes that “plane covering path” in this sense is synonymous with “plane straight-line spanning path” and “non-crossing Hamiltonian path,” and that this differs from covering-path variants in computational geometry where one seeks a geometric path that covers or visits points or objects without necessarily using them as vertices (Aichholzer et al., 2022).

2. Extremal bounds for planar point sets

The modern quantitative theory begins with universal upper and lower bounds for arbitrary planar point sets. Dumitrescu, Gerbner, Keszegh, and Tóth showed that every set of $P$8 points in the plane admits a possibly self-crossing covering path with

$P$9

segments, while for non-crossing covering paths they proved that $P$0 straight-line segments suffice for a small constant $P$1, in fact

$P$2

and they exhibited $P$3-element point sets that require at least

$P$4

segments in every such path (Dumitrescu et al., 2013).

Biniaz sharpened these universal non-crossing bounds. For covering paths, the improved upper bound is

$P$5

and for covering trees,

$P$6

(Biniaz, 2023). The same work recalls the 2014 lower bounds

$P$7

so the asymptotic gap remained substantial (Biniaz, 2023).

The current best bound in the supplied corpus for plane covering paths is

$P$8

which improves the previous best-known upper bound of $P$9 and is achieved by a constructive $P\subset \mathbb{R}^2$0-time algorithm (Akitaya et al., 9 Jul 2025). The same source retains the lower-bound context $P\subset \mathbb{R}^2$1, so the exact asymptotics of $P\subset \mathbb{R}^2$2 remain open (Akitaya et al., 9 Jul 2025).

These inequalities establish a central structural fact: allowing Steiner vertices and permitting points to lie in the interior of edges can reduce the segment count well below the $P\subset \mathbb{R}^2$3 edges forced by spanning paths, but non-crossing still imposes a strong combinatorial penalty relative to the crossing-allowed case (Dumitrescu et al., 2013).

3. Constructive methods and algorithmic complexity

The principal constructions are geometric and strongly tied to Erdős–Szekeres-type structure. Dumitrescu, Gerbner, Keszegh, and Tóth repeatedly extract large caps or cups, using the fact that a cup or cap of size $P\subset \mathbb{R}^2$4 is perfect in the sense that it admits a covering path with $P\subset \mathbb{R}^2$5 segments (Dumitrescu et al., 2013). For non-crossing paths, their strip-based method uses a vertical raster so that each slab contains enough points to force an 18-cap or 18-cup; a structural lemma then saves $P\subset \mathbb{R}^2$6 segments within the slab (Dumitrescu et al., 2013).

Biniaz’s improvement to $P\subset \mathbb{R}^2$7 is based on a finer strip decomposition. A key 23-point lemma states that a set of at least 23 points with distinct $P\subset \mathbb{R}^2$8-coordinates in a vertical strip admits a non-crossing covering path with $P\subset \mathbb{R}^2$9 edges whose endpoints are the leftmost and rightmost points (Biniaz, 2023). The proof uses Erdős–Szekeres with $P$0, exploiting the threshold

$P$1

to obtain a 5-cap or 5-cup, then partitions the strip into convex regions and stitches local non-crossing paths into a global one (Biniaz, 2023). This yields an $P$2-time algorithm after a sort by $P$3-coordinate (Biniaz, 2023).

The $P$4 algorithm is more local. After rotating so that no two points share the same $P$5-coordinate, it scans points from left to right (Akitaya et al., 9 Jul 2025). The first iteration scans one point. Thereafter, each intermediate iteration scans 6 or 7 new points and maintains the invariant that all scanned points are covered by a non-crossing path $P$6 with at most $P$7 segments, lying to the left of the vertical line through the current rightmost scanned point (Akitaya et al., 9 Jul 2025). The local update is determined by the combinatorial type of the convex hull of a 5-point subset $P$8, with explicit case analysis for $P$9 (Akitaya et al., 9 Jul 2025). If 6 points are scanned, the update adds 5 segments; if 7 points are scanned, it adds 6 segments, preserving the amortized ratio $P$0 (Akitaya et al., 9 Jul 2025). The final iteration connects at most 6 remaining points by an $P$1-monotone path (Akitaya et al., 9 Jul 2025).

The computational complexity is essentially tight. Computing a non-crossing covering path requires $P$2 time in the algebraic decision-tree model, via a reduction from sorting (Dumitrescu et al., 2013). The later $P$3 upper bounds therefore match this lower bound asymptotically (Biniaz, 2023, Akitaya et al., 9 Jul 2025).

4. Grid and box-constrained variants

For planar grids, several exact theories coexist because the model itself changes. For the vertices of an $P$4 grid, where segments may have arbitrary orientations and Steiner points are allowed, the minimum number of segments in a non-crossing covering path is exactly

$P$5

while in the crossing-allowed case

$P$6

The same formulas hold for covering trees when counted by edges (Keszegh, 2013). This theory is accompanied by explicit constructions: a non-crossing zigzag snake achieves $P$7, and an axis-aligned spiral achieves $P$8 for the square crossing-allowed case (Keszegh, 2013).

A different square-grid model requires stronger visitation constraints. For

$P$9

a covering path $n-1$0 is a directed polygonal chain that visits each node of $n-1$1 exactly once, while a covering trail may visit nodes at least once; edges cannot be repeated and consecutive edges cannot be collinear (Ripà, 2022). In that framework, $n-1$2 for $n-1$3, and there exists $n-1$4 with

$n-1$5

for every $n-1$6 (Ripà, 2022). If $n-1$7, there also exists a covering cycle $n-1$8 with

$n-1$9

whereas for $\lceil n/2\rceil$0 no covering circuit attains $\lceil n/2\rceil$1 (Ripà, 2022). The same paper gives the upper bound

$\lceil n/2\rceil$2

for the minimum total distance travelled to visit all nodes of $\lceil n/2\rceil$3 with a minimum-link trail (Ripà, 2022).

A further variant imposes an axis-aligned bounding-box constraint in arbitrary dimension. The M$\lceil n/2\rceil$4I algorithm constructs an uncrossing covering path inside the minimum axis-aligned bounding box

$\lceil n/2\rceil$5

for

$\lceil n/2\rceil$6

with exactly

$\lceil n/2\rceil$7

links, each of prescribed length $\lceil n/2\rceil$8 (Ripà, 2024). In the planar specialization $\lceil n/2\rceil$9, this yields an uncrossing covering path for $n-1$0 with

$n-1$1

inside $n-1$2 (Ripà, 2024). This model is not equivalent to the unrestricted grid-covering results above; it trades absolute optimality in segment count for a uniform edge-length and bounding-box constraint.

5. Reconfiguration of plane spanning paths

In reconfiguration theory, a plane covering path is a non-crossing Hamiltonian path. For a planar point set $n-1$3 in general position, $n-1$4 is the set of all plane straight-line paths with vertex set $n-1$5, and a flip on $n-1$6 removes one edge and replaces it with another straight-line edge so that the result is again a valid path in $n-1$7 (Aichholzer et al., 2022). The associated flip graph $n-1$8 has vertex set $n-1$9, with adjacency defined by single flips (Aichholzer et al., 2022). The central open question, due to Akl–Islam–Meijer, is whether $\pi(n),$0 is connected for every point set $\pi(n),$1 in general position (Aichholzer et al., 2022).

Three flip types are identified. Type 1 flips reverse a contiguous chunk from one end of the path, Type 2 flips add the closing edge $\pi(n),$2 when it does not cross the removed edge, and Type 3 flips add $\pi(n),$3 when it crosses exactly one edge of the path (Aichholzer et al., 2022). Type 2 flips can be simulated by a sequence of Type 1 flips, and the paper’s proofs use only Type 1 and Type 2 (Aichholzer et al., 2022).

The main reduction shows that full connectivity would follow from connectivity with the first edge fixed. For distinct $\pi(n),$4,

$\pi(n),$5

and the paper proves the implications “Conjecture 3 $\pi(n),$6 Conjecture 2 $\pi(n),$7 Conjecture 1” (Aichholzer et al., 2022). A viable starting edge $\pi(n),$8 exists if and only if either $\pi(n),$9 or $P$00 is an interior point, or $P$01 and $P$02 are consecutive along the convex hull $P$03 (Aichholzer et al., 2022). Fixing more than the first edge is too restrictive: the analogue of the fixed-first-edge conjecture with the first $P$04 vertices fixed fails, and the paper gives a 7-point example where the induced subgraph is disconnected (Aichholzer et al., 2022).

Two point-set classes admit full positive results. If $P$05 is a wheel set of $P$06 points, then the flip graph on $P$07 is connected with diameter at most $P$08 (Aichholzer et al., 2022). If $P$09 is a generalized double circle of $P$10 points, then the flip graph on $P$11 is connected with diameter $P$12 (Aichholzer et al., 2022). The generalized-double-circle proof uses a spine cycle, a weight function based on cyclic distance along the spine, and valid local flips that either increase the number of spine edges or decrease total weight (Aichholzer et al., 2022). Computationally, Conjecture 1 was verified for all sets of $P$13 points in general position, even when restricted to Type 1 flips (Aichholzer et al., 2022).

6. Coverage-planning and other related formulations

In robotics, coverage path planning addresses a different problem class: the objective is to sweep a planar domain, possibly with holes or obstacles, rather than to cover a finite point set by a few straight segments. One intrinsic approach uses holomorphic quadratic differentials on a triangulated domain $P$14, with natural coordinate

$P$15

so that horizontal and vertical trajectories provide globally coherent sweep lines (Lin et al., 2017). For a planar domain with $P$16 boundaries, doubling $P$17 yields a closed surface of genus $P$18, and a Strebel-type differential decomposes the domain into $P$19 simply connected components (Lin et al., 2017). The resulting dual graph is doubled to obtain an Euler cycle, giving a single continuous covering path with interlaced sweeps in each component (Lin et al., 2017).

A second robotics formulation uses boustrophedon cellular decomposition and a Generalized Traveling Salesman Problem. Here a polygonal region with holes is decomposed into monotone cells, each cell contributes multiple sweep candidates, and the global choice of one sweep per cell together with the visitation order is posed as an Equality-GTSP instance (Bähnemann et al., 2019). The cost model is

$P$20

where $P$21 is the obstacle-aware transition time and $P$22 is the time to execute sweep pattern $P$23 (Bähnemann et al., 2019). On 320 synthetic maps, the planner achieved 14% lower path costs than a conventional coverage planner, and boustrophedon decomposition yielded up to 29% improvement over trapezoidal cell decomposition (Bähnemann et al., 2019).

Other nearby literatures use “covering of the plane” in yet another sense. For a covering of the plane by closed unit discs, the shortest path between two points in the doubly covered region is conjectured to have length at most $P$24, while the proved general upper bound is

$P$25

(Roldán-Pensado, 2013). This is a path problem inside a plane covering, not a covering-path problem for point sets.

Taken together, these lines of work show that “plane covering path” is a term family rather than a single formal object. In point-set combinatorics it refers to low-link non-crossing paths that geometrically cover all input points (Akitaya et al., 9 Jul 2025). In reconfiguration it denotes non-crossing Hamiltonian paths and their flip graphs (Aichholzer et al., 2022). In robotics it refers to complete or near-complete sweeps of continuous domains with obstacle-aware motion planning (Lin et al., 2017, Bähnemann et al., 2019). The distinctions are substantive: they change the admissible vertices, the optimization criterion, the role of planarity, and the main open problems.