Finite and Infinitesimal Rigidity with Polyhedral Norms

Abstract

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R d with respect to polyhedral norms (i.e. norms with closed unit ball P, a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R 2 .

Motions of grid-like reflection frameworks

“…At the combinatorial level, the problem of deciding whether a graph can be realized as a forced symmetric or anti-symmetric isostatic reflection framework is considered and complete characterisations are obtained. Overall this article builds on recent work analyzing the rigidity of frameworks in normed linear spaces, with and without symmetry (see for example [6,7,8,9]).…”

“…However, a characterisation in terms of monochrome subgraph decompositions (analogous to the results in Section 3.1) was not given, as it is not clear whether for an arbitrary decomposition of a signed quotient graph into a monochrome spanning unbalanced map graph and a monochrome spanning tree, there always exists a grid-like realisation of the covering graph with reflectional symmetry which respects the given edge colourings. These realisation problems are non-trivial [8,9] and even arise in the non-symmetric situation [6].…”

On Flattenability of Graphs

“…The second graph is realized in 3-dimensions, but by "unfolding it" as shown, we can flatten it into 2-dimensions Next, we consider the implication of the existence of a generic d-flattenable framework. Specifically, we prove two theorems connecting the d-flattenability with independence in the rigidity matroid: we use the notion of rigidity matrix, and consequently regular frameworks and generic rigidity matroid developed by Kitson [21], as well as the equivalence of finite and infinitesimal rigidity using the notion of well-positioned frameworks, which intuitively means that the l p balls of size given by the corresponding edge-lengths centered at the points intersect properly (i.e, the intersection of k (d − 1)-dimensional ball boundaries is of dimension d − k).…”

The Excluded Minors for Isometric Realizability in the Plane

“…The parameters f p (G) are also widely studied in rigidity theory. We refer the interested reader to Kitson [10] and Sitharam and Gao [16] and the references therein.…”