On Flattenability of Graphs

Sitharam, Meera; Willoughby, Joel

doi:10.1007/978-3-319-21362-0_9

Cited by 8 publications

(6 citation statements)

References 27 publications

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“…The concept is best explained using key theorems of the first author in [59,63] discussed in Section 4.…”

Section: Symmetries Within An Active Constraint Region Via Cayley Conmentioning

confidence: 99%

“…Another advantage is that the method is completely unaffected when δ are intervals rather than exact values [59]. A different characterization of inherent Cayley convexity for a graph G on a set F of non-edges as in the above section has been proven also for higher dimensions d [59,64], showing equivalence to a minor closed property of d-flattenability introduced in [65] and also for other, non-Euclidean distances (norms) in [63]. Any realization of H in a normed space can be flattened into d-dimensional normed space (in the same norm) maintaining the same edge distances.…”

Section: Theorem 8 [59]mentioning

confidence: 99%

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Symmetry in Sphere-Based Assembly Configuration Spaces

et al. 2016

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Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring in nature can be modeled geometrically, starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical, as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates that correspond to various types of macrostates can significantly increase efficiency and accuracy, i.e., reduce the notorious complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency, as well as accuracy vs. efficiency trade-offs in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres, with the following goals.(1) We give new, formal definitions of various concepts relevant to the sphere-based assembly setting that occur in previous work and, in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously-developed concepts include, for example: (i) assembly configuration spaces; (ii) stratification of assembly configuration space into configurational regions defined by active constraint graphs; (iii) paths through the configurational regions; and (iv) coarse assembly pathways. (2) We then demonstrate the new symmetry concepts to compute the sizes and numbers of orbits in two example settings appearing in previous work. (3) Finally, we give formal statements of a variety of open problems and challenges using the new conceptual definitions.Keywords: sphere assembly; configuration space; stratification; distance constraints; Cayley geometry; entropy; kinetics; pathways MotivationSupramolecular assembly is prevalent in nature, healthcare and engineering, but poorly understood. The assembly starts with identical copies of structures drawn from a small number of types. Modeling these starting structures as rigid bunches of spheres is well suited to assembly processes driven by so-called short-range or hard sphere interaction potentials.More formally, an input to a computational model of an assembly process is an assembly system consisting of the following:

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“…The concept is best explained using key theorems of the first author in [59,63] discussed in Section 4.…”

Section: Symmetries Within An Active Constraint Region Via Cayley Conmentioning

confidence: 99%

Section: Theorem 8 [59]mentioning

confidence: 99%

Symmetry in Sphere-Based Assembly Configuration Spaces

et al. 2016

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“…Another point of view is to require only a subset of distances to be preserved, which is the perspective we take in this paper. Our methods are mostly graph theoretical, although similar problems have been studied using techniques from rigidity theory [14,20,21].…”

Section: Introductionmentioning

confidence: 99%

Unavoidable minors for graphs with large $\ell_p$-dimension

Fiorini¹,

Huynh²,

Joret³

et al. 2019

Preprint

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A metric graph is a pair (G, d), where G is a graph and d :for all edges vw ∈ E(G). The p-dimension of G is the least integer k such that there exists an isometric embedding of (G, d) in k p for all distance functions d such that (G, d) has an isometric embedding in K p for some K. It is easy to show that p-dimension is a minor-monotone property. In this paper, we characterize the minor-closed graph classes C with bounded p-dimension, for p ∈ {2, ∞}. For p = 2, we give a simple proof that C has bounded 2-dimension if and only if C has bounded treewidth. In this sense, the 2-dimension of a graph is 'tied' to its treewidth.For p = ∞, the situation is completely different. Our main result states that a minorclosed class C has bounded ∞-dimension if and only if C excludes a graph obtained by joining copies of K4 using the 2-sum operation, or excludes a Möbius ladder with one 'horizontal edge' removed.

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“…Let W 4 denote the wheel on 5 vertices and K 4 + e K 4 be the graph obtained by gluing two copies of K 4 along an edge e and then deleting e, see Figure 1. Using techniques from rigidity matroids, Sitharam and Willoughby [17] determined f ∞ (G) for all graphs G with at most 5 vertices, except for W 4 . They conjectured that W 4 is an excluded minor for f ∞ (G) 2, and that W 4 is the only excluded minor for f ∞ (G) 2.…”

Section: Introductionmentioning

confidence: 99%