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Bill Jackson

Publications and source records attributed to Bill Jackson.

At least 19 recordsLinked to original sources
arXiv2026

Rank Contributions of Vertices in Rigidity Matroids of Clique Covered Graphs

The problems of characterizing the graphs $G$ which are generically rigid in ${\mathbb R}^d$, or more generally, determining the rank function of the $d$-dimensional rigidity matroid ${\cal R}_d(G)$ of an arbitrary graph $G$, have been solved when $d\leq 2$ but are major open problems in discrete geometry when $d\geq 3$. In this paper we shall concentrate on the case when $d=3$. We first revisit a conjecture of Dress from 1987 that the rank of the ${\cal R}_3$-closure of a graph $G$ is determined by its maximal complete subgraphs of size at least five. We show that his conjectured value for the rank of the closure gives an upper bound on the actual value. We also deduce that the truth of this conjecture would imply a good characterization of the rank of ${\cal R}_3(G)$ for all graphs $G$. The rank formula in Dress's conjecture leads us to consider the family of $K_t$-covered graphs, i.e., graphs in which every edge belongs to a complete subgraph $K_t$, for some $t\geq 3$. This family contains several well-studied graph classes such as body-pin graphs, combinatorial zeolites, and molecular graphs. We introduce a new notion of rank contributions of vertices in an arbitrary matroid on the edge set of a graph $G$, and use it to obtain lower bounds on the rank contributions of vertices in ${\cal R}_3(G)$ and ${\cal C}^1_2(G)$ when $G$ is $K_t$-covered. We use these bounds to show that a conjectured min-max formula for the rank of body-pin graphs in ${\cal R}_3$ holds for the $C_2^1$-cofactor matroid (which is conjectured by Whiteley to be equal to ${\cal R}_3$), and to obtain new sufficient connectivity conditions for the (global) rigidity of $K_4$- and $K_5$-covered graphs in ${\mathbb R}^3$.

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arXiv2026

Symmetric Powers of Matroids

The study of matroid products has become an active area of research, owing to their connections with tropical ideals and linear representability. In this paper, we study matroidal abstractions of the multilinearity of symmetric powers of vector spaces, using a duality between symmetric powers of matroids and abstract rigidity. These observations allow us to solve Mason's conjecture concerning the equivalence of two definitions of a symmetric power of a matroid. We show that Mason's conjecture holds for second symmetric powers of matroids whereas it fails for third symmetric powers.

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arXiv2025

Sufficient conditions for bipartite rigidity, symmetric completability and hyperconnectivity of graphs

We consider three matroids defined by Kalai in 1985: the symmetric completion matroid $\mathcal{S}_d$ on the edge set of a looped complete graph; the hyperconnectivity matroid $\mathcal{H}_d$ on the edge set of a complete graph; and the birigidity matroid $\mathcal{B}_d$ on the edge set of a complete bipartite graph. These matroids arise in the study of low rank completion of partially filled symmetric, skew-symmetric and rectangular matrices, respectively. We give sufficient conditions for a graph $G$ to have maximum possible rank in these matroids. For $\mathcal{S}_d$ and $\mathcal{H}_d$, our conditions are in terms of the minimum degree of $G$ and are best possible. For $\mathcal{B}_d$, our condition is in terms of the connectivity of $G$. Our results have several implications for the unique completability of low-rank matrices. In particular, they imply that: almost all sufficiently large $n \times n$ positive semidefinite matrices of rank $d$ are uniquely determined by any subset of their entries which includes at least $(n + d + 1)/2$ entries from each row; almost all $m \times n$ matrices of rank $d$ are uniquely determined by any subset of their entries whose positions define a spanning subgraph of $K_{m,n}$ which is $k_d$-connected, for some constant $k_d=\mbox{O}(d^3)$.

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arXiv2025

Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory

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.

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arXiv2025

$k$-fold circuits and coning in rigidity matroids

In 1980 Lov\'{a}sz introduced the concept of a double circuit in a matroid. The 2nd, 3rd and 4th authors recently generalised this notion to $k$-fold circuits (for any natural number $k$) and proved foundational results about these $k$-fold circuits. In this article we use $k$-fold circuits to derive new results on the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. These results include analysing 2-sums, showing sufficient conditions for the $k$-fold circuit property to hold for $k$-fold $\mathcal{R}_d$-circuits, and giving an extension of Whiteley's coning lemma. The last of these allows us to reduce the problem of determining if a graph $G$ with a vertex $v$ of sufficiently high degree is independent in $\mathcal{R}_d$ to that of verifying matroidal properties of $G-v$ in $\mathcal{R}_{d-1}$.

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arXiv2025

Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach

A $d$-dimensional (bar-and-joint) framework $(G,p)$ consists of a graph $G=(V,E)$ and a realisation $p:V\to \mathbb{R}^d$. It is rigid if every continuous motion of the vertices which preserves the lengths of the edges is induced by an isometry of $\mathbb{R}^d$. The study of rigid frameworks has increased rapidly since the 1970s stimulated by numerous applications in areas such as civil and mechanical engineering, CAD, molecular conformation, sensor network localisation and low rank matrix completion. We will describe some of the main results in combinatorial rigidity theory and their applications to other areas of combinatorics, putting an emphasis on links to matroid theory.

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arXiv2025

Symmetric Tensor Matroids, Dual Rigidity Matroids, and the Maximality Conjecture

Inspired by a recent result of Brakensiek et al. that symmetric tensor matroids and rigidity matroids are linked by matroid duality, we define abstract symmetric tensor matroids as a dual concept to abstract rigidity matroids and establish their basic properties. We then exploit this duality to obtain an alternative characterisation of the generic $d$-dimensional rigidity on $K_n$ for $n-d\leq 6$ to that given by Grasseger et al. Our results imply that Graver's maximality conjecture holds for these matroids. We also consider the related family of $K_{1,t+1}$-matroids on $K_n$ and show that this family has a unique maximal element only when $t\leq 3$. This implies that the family of second quasi symmetric powers of the uniform matroid $U_{t,n}$ does not have a unique maximal matroid if $t\geq 4$ and $n$ is sufficiently large.

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arXiv2025

Volume Rigidity of Simplicial Manifolds

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex simplicial polyhedron $P$ is rigid i.e. every continuous motion of the vertices of $P$ in $\mathbb R^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex smplicial polytopes in $\mathbb R^d$ for all $d\geq 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in $\mathbb R^{d}$ by Kalai for $d\geq 4$ and Fogelsanger for $d\geq 3$. We will generalise Kalai's result by showing that, for all $d\geq 4$ and any fixed $1\leq k\leq d-3$, every generic realisation of the $k$-skeleton of a simplicial $(d-1)$-manifold in $\mathbb R^{d}$ is volume rigid, i.e. every continuous motion of its vertices in $\mathbb R^d$ which preserves the volumes of its $k$-faces results in a congruent realisation. In addition, we conjecture that our result remains true for $k=d-2$ and verify this conjecture when $d=4,5,6$.

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arXiv2024

The $k$-fold circuit property for matroids

Double circuits were introduced by Lov\'{a}sz in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in linear matroids. This formula was extended to all matroids satisfying the so-called `double circuit property' by Dress and Lov\'{a}sz in 1987. We extend these notions to $k$-fold circuits for all natural numbers $k$ and show, in particular that several families of matroids which are known to satisfy the double circuit property, satisfy the $k$-fold circuit property for all natural numbers $k$. These families include all pseudomodular matroids (such as full linear, algebraic and transversal matroids) and certain families of count matroids. These results suggest that the $k$-fold circuit property can be used as a measure of how close the lattice of flats of a matroid is to being a modular lattice.

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arXiv2024

Globally Rigid Convex Braced Polygons

Here we propose a class of frameworks in the plane, braced polygons, that may be globally rigid and are analogous to convex polyopes in 3 space that are rigid by Cauchy's rigidity Theorem in 1813.

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arXiv2023

Rigidity of Symmetric Simplicial Complexes and the Lower Bound Theorem

We show that, if $\Gamma$ is a point group of $\mathbb{R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal S$ is a $k$-pseudomanifold which has a free automorphism of order two, then either $\mathcal S$ has a $\Gamma$-symmetric infinitesimally rigid realisation in $\mathbb{R}^{k+1}$ or $k=2$ and $\Gamma$ is a half-turn rotation group.This verifies a conjecture made by Klee, Nevo, Novik and Zhang for the case when $\Gamma$ is a point-inversion group. Our result implies that Stanley's lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial involution, thus verifying (the inequality part) of another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes.

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arXiv2022

Global Rigidity of Triangulated Manifolds

We prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case $d=3$ verifies a conjecture of Connelly. Our results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. We also give two significant applications of our main theorems. We show that that the characterisation of pseudomanifolds with extremal edge numbers given by the Lower Bound Theorem extends to circuits of the simplicial matroid. We also prove the generic case of a conjecture of Kalai concerning the reconstructability of a polytope from its space of stresses. The proofs of our main results adapt earlier ideas of Fogelsanger and Whiteley to the setting of global rigidity. In particular we verify a special case of Whiteley's vertex splitting conjecture for global rigidity.

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arXiv2021

Coincident Rigidity of 2-Dimensional Frameworks

Fekete, Jord\'an and Kaszanitzky [4] characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their characterisation to an arbitrary set T of vertices and verify our conjecture when |T| = 3.

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arXiv2021

Maximal Matroids in Weak Order Posets

Let $\cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $\cX$-matroid if each set in $\cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $\cX$-matroid in the weak order poset of all $\cX$-matroids on $E$, and characterizing its rank function when it exists.

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arXiv2020

An improved bound for the rigidity of linearly constrained frameworks

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Theran in 2010 and the case when each vertex is constrained to lie in an affine subspace of dimension $t$, and $d\geq t(t-1)$ was solved by Cruickshank, Guler and the first two authors in 2019. We extend the latter result by showing that the given characterisation holds whenever $d\geq 2t$.

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arXiv2020

Flexible circuits in the $d$-dimensional rigidity matroid

A bar-joint framework $(G,p)$ in $\mathbb{R}^d$ is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of $\mathbb{R}^d$. It is known that, when $(G,p)$ is generic, its rigidity depends only on the underlying graph $G$, and is determined by the rank of the edge set of $G$ in the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. Complete combinatorial descriptions of the rank function of this matroid are known when $d=1,2$, and imply that all circuits in $\mathcal{R}_d$ are generically rigid in $\mathbb{R}^d$ when $d=1,2$. Determining the rank function of $\mathcal{R}_d$ is a long standing open problem when $d\geq 3$, and the existence of non-rigid circuits in $\mathcal{R}_d$ for $d\geq 3$ is a major contributing factor to why this problem is so difficult. We begin a study of non-rigid circuits by characterising the non-rigid circuits in $\mathcal{R}_d$ which have at most $d+6$ vertices.

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arXiv2019

Abstract 3-Rigidity and Bivariate $C_2^1$-Splines I: Whiteley's Maximality Conjecture

A conjecture of Graver from 1991 states that the generic $3$-dimensional rigidity matroid is the unique maximal abstract $3$-rigidity matroid with respect to the weak order on matroids. Based on a close similarity between the generic $d$-dimensional rigidity matroid and the generic $C_{d-2}^{d-1}$-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic $C_{d-2}^{d-1}$-cofactor matroid is the unique maximal abstract $d$-rigidity matroid for all $d\geq 2$. We verify the case $d=3$ of Whiteley's conjecture in this paper. A key step in our proof is to verify a second conjecture of Whiteley that the `double V-replacement operation' preserves independence in the generic $C_2^1$-cofactor matroid.

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