Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918134992076800 |
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| author | Garamvölgyi, Dániel Jackson, Bill Jordán, Tibor |
| author_facet | Garamvölgyi, Dániel Jackson, Bill Jordán, Tibor |
| contents | A graph is $\mathcal{R}_d$-independent (resp. $\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\mathcal{R}_d$-independent graph satisfies the sparsity condition $|E(H)|\leq d|V(H)|-\binom{d+1}{2}$ for all subgraphs $H$ with at least $d+1$ vertices. Several other families of graphs $G$ arising naturally in rigidity theory, such as minimally globally $d$-rigid graphs, are known to satisfy the bound $|E(G)|\leq (d+1)|V(G)|-\binom{d+2}{2}$. We unify and extend these results by considering the family of $d$-stress-independent graphs which includes many of these families. We show that every $d$-stress-independent graph is $\mathcal{R}_{d+1}$-independent. A key ingredient in our proofs is the concept of $d$-stress-linked pairs of vertices. We derive a new sufficient condition for $d$-stress linkedness and use it to obtain a similar condition for a pair of vertices of a graph to be globally $d$-linked. This result strengthens a result of Tanigawa on globally $d$-rigid graphs. We also show that every minimally $\mathcal{R}_d$-connected graph $G$ is $\mathcal{R}_{d+1}$-independent and that the only subgraphs of $G$ that can satisfy Maxwell's criterion for $\mathcal{R}_{d+1}$-independence with equality are copies of $K_{d+2}$. Our results give affirmative answers to two conjectures in graph rigidity theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory Garamvölgyi, Dániel Jackson, Bill Jordán, Tibor Combinatorics Metric Geometry A graph is $\mathcal{R}_d$-independent (resp. $\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\mathcal{R}_d$-independent graph satisfies the sparsity condition $|E(H)|\leq d|V(H)|-\binom{d+1}{2}$ for all subgraphs $H$ with at least $d+1$ vertices. Several other families of graphs $G$ arising naturally in rigidity theory, such as minimally globally $d$-rigid graphs, are known to satisfy the bound $|E(G)|\leq (d+1)|V(G)|-\binom{d+2}{2}$. We unify and extend these results by considering the family of $d$-stress-independent graphs which includes many of these families. We show that every $d$-stress-independent graph is $\mathcal{R}_{d+1}$-independent. A key ingredient in our proofs is the concept of $d$-stress-linked pairs of vertices. We derive a new sufficient condition for $d$-stress linkedness and use it to obtain a similar condition for a pair of vertices of a graph to be globally $d$-linked. This result strengthens a result of Tanigawa on globally $d$-rigid graphs. We also show that every minimally $\mathcal{R}_d$-connected graph $G$ is $\mathcal{R}_{d+1}$-independent and that the only subgraphs of $G$ that can satisfy Maxwell's criterion for $\mathcal{R}_{d+1}$-independence with equality are copies of $K_{d+2}$. Our results give affirmative answers to two conjectures in graph rigidity theory. |
| title | Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2509.03150 |