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On generic $3$-rigidity of graphs

Tamás Baranyai

TL;DR

This work addresses the problem of characterizing generic $3$-rigidity of graphs by reducing it to planar subproblems through a three-way edge partition. It introduces a precise decomposition criterion: a graph is minimally $3$-rigid if and only if for every edge $e$ there exist edge-sets $S_1,S_2,S_3$ with $|S_i|=|V|-i$ such that $G(S_1\cup S_2)$, $G(S_1\cup S_3)/e$, and $G(S_2\cup S_3)/e$ are minimally $2$-rigid. The methodology hinges on the $3$-D rigidity matrix $M(G,3)$ and its reduced form $N(G,3)$, analyzing planarity-inspired block structures to connect $3$-rigidity with planar $2$-rigidity tests and edge-contraction techniques. The result provides an exact combinatorial criterion and a pathway toward efficient algorithms for verifying $3$-dimensional rigidity in graphs.

Abstract

We aim to give an exact condition of generic $3$ -rigidity of graphs relying on partitioning the edges into $3$ subsets; such that each subset-pair gives a generically $2$-rigid graph, either by themselves or after an appropriate edge-deletion.

On generic $3$-rigidity of graphs

TL;DR

This work addresses the problem of characterizing generic -rigidity of graphs by reducing it to planar subproblems through a three-way edge partition. It introduces a precise decomposition criterion: a graph is minimally -rigid if and only if for every edge there exist edge-sets with such that , , and are minimally -rigid. The methodology hinges on the -D rigidity matrix and its reduced form , analyzing planarity-inspired block structures to connect -rigidity with planar -rigidity tests and edge-contraction techniques. The result provides an exact combinatorial criterion and a pathway toward efficient algorithms for verifying -dimensional rigidity in graphs.

Abstract

We aim to give an exact condition of generic -rigidity of graphs relying on partitioning the edges into subsets; such that each subset-pair gives a generically -rigid graph, either by themselves or after an appropriate edge-deletion.
Paper Structure (3 sections, 4 theorems, 7 equations)

This paper contains 3 sections, 4 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Self-stresses of frameworks
  3. $3$-rigidity of graphs

Key Result

Lemma 1

For any $e\in E$ and any coordinate plane $[\boldsymbol{x}_k,\boldsymbol{x}_l ]$ if $\boldsymbol{B}\boldsymbol{s}^k=\boldsymbol{0}$ and $\boldsymbol{B}\boldsymbol{s}^l=\boldsymbol{0}$ are known to hold and the directional constraints in the plane are known to be satisfied for all edges except for $e

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof