Connected rigidity matroids and unique realizations of graphs

Bill Jackson, T. Jordán

Research output: Contribution to journalArticle

215 Citations (Scopus)

Abstract

A d-dimensional framework is a straight line realization of a graph G in ℝd. We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in ℝd if every equivalent framework can be obtained from it by an isometry of ℝd. Bruce Hendrickson proved that if G has a unique realization in ℝd then G is (d + 1)-connected and redundantly rigid. He conjectured that every realization of a (d + 1)-connected and redundantly rigid graph in ℝd is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson's conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a two-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected.

Original languageEnglish
Pages (from-to)1-29
Number of pages29
JournalJournal of Combinatorial Theory. Series B
Volume94
Issue number1
DOIs
Publication statusPublished - May 2005

Fingerprint

Matroid
Rigidity
Graph in graph theory
Connected graph
Framework
Isometry
Straight Line
Deduce
Resolve
Corollary

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Connected rigidity matroids and unique realizations of graphs. / Jackson, Bill; Jordán, T.

In: Journal of Combinatorial Theory. Series B, Vol. 94, No. 1, 05.2005, p. 1-29.

Research output: Contribution to journalArticle

@article{fd246773eeae4b408c7611a088624603,
title = "Connected rigidity matroids and unique realizations of graphs",
abstract = "A d-dimensional framework is a straight line realization of a graph G in ℝd. We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in ℝd if every equivalent framework can be obtained from it by an isometry of ℝd. Bruce Hendrickson proved that if G has a unique realization in ℝd then G is (d + 1)-connected and redundantly rigid. He conjectured that every realization of a (d + 1)-connected and redundantly rigid graph in ℝd is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson's conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a two-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected.",
author = "Bill Jackson and T. Jord{\'a}n",
year = "2005",
month = "5",
doi = "10.1016/j.jctb.2004.11.002",
language = "English",
volume = "94",
pages = "1--29",
journal = "Journal of Combinatorial Theory. Series B",
issn = "0095-8956",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Connected rigidity matroids and unique realizations of graphs

AU - Jackson, Bill

AU - Jordán, T.

PY - 2005/5

Y1 - 2005/5

N2 - A d-dimensional framework is a straight line realization of a graph G in ℝd. We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in ℝd if every equivalent framework can be obtained from it by an isometry of ℝd. Bruce Hendrickson proved that if G has a unique realization in ℝd then G is (d + 1)-connected and redundantly rigid. He conjectured that every realization of a (d + 1)-connected and redundantly rigid graph in ℝd is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson's conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a two-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected.

AB - A d-dimensional framework is a straight line realization of a graph G in ℝd. We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in ℝd if every equivalent framework can be obtained from it by an isometry of ℝd. Bruce Hendrickson proved that if G has a unique realization in ℝd then G is (d + 1)-connected and redundantly rigid. He conjectured that every realization of a (d + 1)-connected and redundantly rigid graph in ℝd is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson's conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a two-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected.

UR - http://www.scopus.com/inward/record.url?scp=15444370051&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15444370051&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2004.11.002

DO - 10.1016/j.jctb.2004.11.002

M3 - Article

VL - 94

SP - 1

EP - 29

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -