Rigid components in fixed-lattice and cone frameworks

Rigid components in fixed-lattice and cone frameworks Matthew Berardi∗ Brent Heeringa† Justin Malestein∗ Louis Theran∗ Abstract We study the fundamental algorithmic rigidity problems for generic frameworks periodic with respect to a fixed lattice or a finite-order rotation in the plane. For fixed- lattice frameworks we give an O(n2) algorithm for de- ciding generic rigidity and an O(n3) algorithm for com- puting rigid components. If the order of rotation is part of the input, we give an O(n4) algorithm for deciding rigidity; in the case where the rotation’s order is 3, a more specialized algorithm solves all the fundamental algorithmic rigidity problems in O(n2) time. 1 Introduction The geometric setting for this paper involves two varia- tions on the well-studied planar bar-joint rigidity model: fixed-lattice periodic frameworks and cone frameworks. A fixed-lattice periodic framework is an infinite struc- ture, periodic with respect to a lattice, where the al- lowed continuous motions preserve, the lengths and con- nectivity of the bars, as well as the periodicity with re- spect to a fixed lattice. See Figure 1(a) for an example. A cone framework is also made of fixed-length bars con- nected by universal joints, but it is finite and symmet- ric with respect to a finite order rotation; the allowed continuous motions preserve the bars’ lengths and con- nectivity and symmetry with respect to a fixed rotation center. Cone frameworks get their name from the fact that the quotient of the plane by a finite order rotation is a flat cone with opening angle 2π/k and the quotient framework, embedded in the cone with geodesic “bars”, captures all the geometric information [12]. Figure 2(a) shows an example. A fixed-lattice framework is rigid if the only allowed motions are translations and flexible otherwise. A cone- framework is rigid if the only allowed motions are rota- tions around the center and flexible otherwise. The al- ternate formulation for cone frameworks says that rigid- ity means the only allowed motions are isometries of the cone, which is just rotation around the cone point. A framework is minimally rigid if it is rigid, but ceases to be so if any of the bars are removed. ∗Department of Mathematics, Temple University, {mberardi,justmale,theran}@temple.edu †Department of Computer Science, Williams College, heeringa@cs.williams.edu Generic rigidity The combinatorial model for the fixed-lattice and cone frameworks introduced above is given by a colored graph (G, γ): G = (V, E) is a finite directed graph and γ = (γij)ij∈E is an assignment of a group element γij ∈Γ (the “color”) to each edge ij for a group Γ. For fixed-lattice frameworks, the group Γ is Z2, representing translations; for cone frameworks it is Z/kZ with k ≥2 a natural number. See Figure 1(b) and Figure 2(b). The colors can be seen as efficiently encoding a map ρ from the oriented cycle space of G into Γ; ρ is defined, in detail, in Section 2. If the image of ρ restricted to a sub- graph G′ contains only the identity element, we define the Γ-image of ρ to be trivial otherwise it is non-trivial. The generic rigidity theory of planar frameworks with, (a) (0,1) (0,-1) (1,0) (0,0) (0,0) (0,0) (b) Figure 1: Periodic frameworks and colored graphs: (a) part of a periodic framework, with the representation of the integer lattice Z2 shown in gray and the bars shown in black; (b) one possibility for the the associated colored graph with Z2 colors on the edges. (Graphics from [11].) more generally, crystallographic symmetry has seen a lot of progress recently [3, 11, 12, 14]. Elissa Ross [14] announced the following theorem: Theorem 1 ([11, 14]) A generic fixed-lattice periodic framework with associated colored graph (G, γ) is min- imally rigid if and only if: (1) G has n vertices and 2n −2 edges; (2) all non-empty subgraphs G′ of G with m′ edges and n′ vertices and trivial Z2-image satisfy m′ ≤2n′ −3; (3) all non-empty subgraphs G′ with non- trivial Z2-image satisfy m′ ≤2n′ −2. The colored graphs appearing in the statement of The- orem 1 are defined to be Ross graphs; if only condi- arXiv:1105.3234v1 [cs.DS] 16 May 2011 (a) 1 1 (b) (c) Figure 2: Cone-Laman graphs: (a) a realization of the framework on a cone with opening angle 2π/3 (graphic from Chris Thompson); (b) a Z/3Z-colored graph (edges without colors have color 0); (c) the developed graph with Z/3Z-symmetry (dashed edges are lifts of dashed edges in (b)). tions (2) and (3) are met, (G, γ) is Ross-sparse. Ross graphs generalize the well-known Laman graphs which are uncolored, have m = 2n −3 edges, and satisfy (2). By Theorem 1 the maximal rigid sub-frameworks of a generic fixed-lattice framework on a Ross-sparse colored graph (G, γ) correspond to maximal subgraphs of G with m′ = 2n′ −2; we define these to be the rigid com- ponents of (G, γ). In the sequel, we will also refer to graphs with the Ross property for Γ = Z/kZ as simply “Ross graphs”. Malestein and Theran [12] proved a similar statement for cone frameworks: The

This content is AI-processed based on open access ArXiv data.