{
  "id": "1010.5552",
  "version": "v2",
  "published": "2010-10-27T02:17:49.000Z",
  "updated": "2012-09-13T20:05:10.000Z",
  "title": "Directed Graphs, Decompositions, and Spatial Linkages",
  "authors": [
    "Offer Shai",
    "Adnan Sljoka",
    "Walter Whiteley"
  ],
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "The decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned d-isostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs), or equivalently into minimal pinned isostatic graphs, which we call d-Assur graphs. We also study key properties of motions induced by removing an edge in a d-Assur graph - defining a stronger sub-class of strongly d-Assur graphs by the property that all inner vertices go into motion, for each removed edge. The strongly 3-Assur graphs are the central building blocks for kinematic linkages in 3-space and the 3-Assur graphs are components in the analysis of built linkages. The d-Assur graphs share a number of key combinatorial and geometric properties with the 2-Assur graphs, including an associated lower block- triangular decomposition of the pinned rigidity matrix which provides modular information for extending the motion induced by inserting one driver in a bottom Assur linkage to the joints of the entire linkage. We also highlight some problems in combinatorial rigidity in higher dimensions (d > 2) which cause the distinction between d-Assur and strongly d-Assur which did not occur in the plane.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-13T20:05:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C25",
      "70B15",
      "05C20",
      "05C85"
    ],
    "keywords": [
      "directed graphs",
      "spatial linkages",
      "decomposition",
      "minimal pinned isostatic graphs",
      "d-assur graphs share"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.5552S"
    }
  }
}