Abstract-Path Unfolding Algorithm

• Abstract-Path Unfolding is a family of combinatorial and geometric methodologies that unfold convex polyhedral surfaces into non-overlapping planar layouts by constructing specific cutting paths.
• The approach employs techniques such as greedy radially monotone forests, dual Hamiltonian paths, and chain code tracking to ensure accurate and efficient planar developments.
• Empirical studies demonstrate robust performance, with over 95% success in generating non-overlapping unfoldings for spherical polyhedra and extensions to orthotubes and cubes.

The Abstract-Path-Unfolding algorithm is a family of combinatorial and geometric methodologies for unfolding convex polyhedral surfaces and combinatorial polyhedral objects into non-overlapping planar layouts. The central theme of these algorithms is the construction of paths (cut-paths or dual paths) that encode how to develop the surface—typically by exploiting monotonicity, Hamiltonicity, or quasigeodesic properties—so that the resulting planar figure avoids overlap. Specific instantiations have appeared for convex polyhedra, orthotubes, higher-dimensional cubes, and as combinatorial analogues in partial-order reduction in concurrency theory. The following sections survey key definitions, algorithmic structures, geometric guarantees, complexity bounds, and representative applications.

1. Foundational Definitions and Structural Principles

In the context of convex polyhedral geometry, an abstract-path-unfolding is characterized by a selection of surface paths that, when cut, allow the polyhedron to be developed isometrically into the plane. For example, radially monotone paths on a triangulated convex domain $C\subset\mathbb R^2$ are defined such that for any vertex $v$, the directed path $Q=(v_0,\ldots,v_k)$ is radially monotone with respect to $v_i$ if the distance $d(p,v_i)=\|p-v_i\|$ is non-decreasing as $Q$ is traversed outward from $v_i$. Equivalently, $Q$ crosses each concentric circle centered at $v_i$ at most once, or, via angle test, all subsequent vertex-turns after $v_i$ satisfy $\theta_j(v_i)\ge\pi/2$.

For cut trees on polyhedral surfaces, a radially monotone cut tree $T$ is a spanning tree such that for each leaf $\ell$, the unique root path $\ell\rightarrow r$ is a radially monotone path (when developed to the plane).

In the combinatorial setting—e.g., in unfolding cubes or orthotubes—the surface graph is represented as a planar adjacency graph, and paths (spanning forests, Hamiltonian paths) are constructed in the dual.

2. Algorithmic Constructions

2.1 Greedy Radially Monotone Spanning Forest (2D)

On convex, non-obtusely triangulated domains, the existence of a radially monotone spanning forest (rm-forest) is guaranteed. The construction proceeds as follows:

• Sort interior vertices by increasing distance to the center of the minimal enclosing circle.
• For each interior vertex, connect to an admissible neighbor so that the candidate path, if added, preserves radial monotonicity with respect to the current vertex.
• The connection with the smallest worst turn-angle is favored.

Key structural ingredient: hourglass paths, wherein each edge lies within a strict 90° in-cone or out-cone at each step, allow extension without loss of radial monotonicity.

2.2 Extension to 3D Spherical Polyhedra

For convex polyhedra inscribed in a sphere, the greedy approach is mimicked by sorting vertices by decreasing latitude, connecting each to an admissible "parent" below it, and verifying that the medial chain (constructed from face angles and curvatures) remains radially monotone upon development to the plane. The algorithm tends to succeed in practice, subject to triangulation constraints.

2.3 Dual Hamiltonian Path Unfolding of Orthotubes

For orthotubes (sequences of unit cubes joined face-to-face), the surface is represented as a planar dual graph whose vertices are surface (rectangular) faces. The algorithm synthesizes a Hamiltonian path in this dual graph, expressed via a "chain code" over $\{L, R, S\}$ (left, right, straight). The chain code is constructed so that the prefix turn accumulator $qturn(U)$ remains bounded ($|qturn|\leq 1$), guaranteeing monotonicity and non-overlap, and every outer face is visited exactly once.

Pseudocode excerpts detail construction and extension logic, handling "local patterns" (straight, left, right, opposite) and maintaining prefix invariants for monotonic planar development.

2.4 Combinatorial Frameworks for Cube Unfolding

Unfolding the $n$-cube is formalized by constructing Hamiltonian paths in the $n$-Roberts (cocktail-party) graph, with geometric development in $\mathbb R^{n-1}$. Combinatorial properties relate cycles and paths in this graph to chord diagrams and integer partitions, allowing explicit characterization of bounding boxes of unfolded structures.

3. Geometric and Combinatorial Guarantees

Central to all abstract-path-unfolding schemes are guarantees that the selected path/cut structure yields overlap-free planar layouts:

• Radially Monotone Guarantee: If a path is radially monotone with respect to all its vertices, opening along curvature ensures that the left and right developments do not cross (cf. Theorem 3.1, (O'Rourke, 2016)).
• Prefix-Turn Bound: In orthotube unfolding, if at every prefix the accumulated turn is at most one, the corresponding lattice path in the plane is monotone (never reverses direction), so rows of unit squares are always stacked either above or below previous ones, ensuring non-overlap (cf. Lemma 1, (Demaine et al., 2022)).
• Quasigeodesic Unfolding: For star unfoldings along a simple quasigeodesic loop $Q$, shortest geodesics from each vertex to $Q$ split the polyhedron into convex or one-corner-convex planar disks. Gluing along a supporting boundary segment ensures non-intersection (0812.2257).
• Cube Unfoldings: Chord-diagram conditions and strict monotonicity (no u-turns along coordinate directions) guarantee that all facets are embedded disjointly in the bounding box.

4. Complexity, Empirical Performance, and Implementation Aspects

Algorithmic complexity for geometric path-unfolding reflects graph traversal and domain tests:

• 2D Radially Monotone Forest: $O(n^2)$ time, with sorting in $O(n \log n)$ and path-checking over $O(n)$ vertices (O'Rourke, 2016).
• 3D Spherical Polyhedron: Also $O(n^2)$, efficient in practice due to bounded vertex degree.
• Orthotube Dual Hamiltonian Path: All steps are $O(n)$, with local pattern recognition and continuability tested in constant time per step (Demaine et al., 2022).
• Quasigeodesic Star Unfolding: Multi-source shortest paths in $O((n+q)\log(n+q))$ time, using established geodesic graph algorithms (0812.2257).

Empirical studies confirm that radially monotone trees are found in over 95% of random spherical polyhedra tested; local corrections (splitting obtuse triangles) further increase the success rate (O'Rourke, 2016). Similar robustness is reported for orthotube unfolding and cube developments.

Implementation recommendations include robust numeric tests, careful data structures (half-edge/winged-edge), explicit tracking of chain codes, and geometric validation steps.

5. Classification, Correspondence, and Theoretical Extensions

The abstract-path-unfolding paradigm enables deeper combinatorial classification:

• Integer Partition Classification: For cube unfoldings, bounding box dimensions correspond to $n$-cube partitions of $3n-2$, and every such partition can be realized by a path unfolding (DeSplinter et al., 2020).
• Chord Diagram Bijections: Hamiltonian cycles and paths in Roberts graphs correspond to specific chord diagrams (loopless or one-loop) on $2n$ vertices, offering a combinatorial invariant for the unfolding structure.

Open theoretical questions involve the universality of radially monotone cut trees for arbitrary triangulations, extension to higher genus or non-convex surfaces, and optimization for minimal cut length or total turn.

6. Limitations, Extensions, and Connections to Other Domains

Algorithmic limitations include the dependence on triangulation properties: success is reduced in configurations with obtuse angles, which may necessitate local mesh refinement. The framework has been extended—empirically and conjecturally—to random spherical polyhedra and orthotube shapes, with robust performance.

Research flags future work in proving existence theorems, generalizing to self-approaching curves, and extending to complex surface topologies. In concurrency, abstract path-unfolding techniques have inspired state-space reduction strategies and have combinatorial parallels to Mazurkiewicz trace theory and partial-order reasoning.

A plausible implication is that the abstract-path-unfolding methodology serves as a unifying conceptual framework across geometric unfolding and combinatorial state-space analysis, providing both concrete algorithms and structural insights into the underlying spaces of cuts, paths, and developments.