ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.10
https://doi.org/10.26493/1855-3974.2826.3dc
(Also available at http://amc-journal.eu)
Coincident-point rigidity in normed planes
Sean Dewar *
School of Mathematics, University of Bristol, Bristol BS8 1UG, U.K
John Hewetson
Dept. Math. Stats., Lancaster University, Lancaster LA1 4YF, U.K
Anthony Nixon †
Dept. Math. Stats., Lancaster University, Lancaster LA1 4YF, U.K
Received 11 February 2022, accepted 28 March 2023, published online 7 September 2023
Abstract
A bar-joint framework (G, p) is the combination of a graph G and a map p assigning
positions, in some space, to the vertices of G. The framework is rigid if every edge-length-
preserving continuous motion of the vertices arises from an isometry of the space. We
will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated
vertices are mapped to the same position. This non-genericity assumption leads us to a
count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that
independence in this matroid is equivalent to independence as a suitably regular bar-joint
framework in a non-Euclidean normed plane with two coincident points; this characterises
when a regular non-Euclidean normed plane coincident-point framework is rigid and allows
us to deduce a delete-contract characterisation. We then apply this result to show that
an important construction operation (generalised vertex splitting) preserves the stronger
property of global rigidity in non-Euclidean normed planes and use this to construct rich
families of globally rigid graphs when the non-Euclidean normed plane is analytic.
Keywords: Bar-joint framework, global rigidity, non-Euclidean framework, count matroid, recursive
construction, normed spaces, analytic norm.
Math. Subj. Class. (2020): 52C25, 05C10, 52B40, 46B20
*Corresponding author. Supported in part by the Austrian Science Fund (FWF): P31888 and in part by the
Heilbronn Institute for Mathematical Research.
†Supported in part by the Heilbronn Institute for Mathematical Research and in part by EPSRC grant number
EP/W019698/1.
E-mail addresses: sean.dewar@bristol.ac.uk (Sean Dewar), john.hewetson02@gmail.com (John Hewetson),
a.nixon@lancaster.ac.uk (Anthony Nixon)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P1.10
1 Introduction
A bar-joint framework (G, p) is the combination of a graph G = (V,E) and a map p : V →
Rd assigning positions to the vertices of G (and hence lengths to the edges). Note that in
this article graphs are taken to be finite and simple. Intuitively, the framework is rigid if
every edge-length-preserving continuous motion of the vertices arises from an isometry of
Rd. More strongly, (G, p) is globally rigid if every framework in Rd, on the same graph,
with the same edge lengths actually has the same distance between every pair of vertices.
The rigidity and global rigidity of bar-joint frameworks in Euclidean spaces has been
intensely studied in recent years (e.g. [2, 3, 13, 16, 21, 23]) and has a rich history going
as far back as classical work of Euler and Cauchy on Euclidean polyhedra. In the last
decade, work on rigidity has been generalised to various non-Euclidean normed spaces
(e.g. [6, 9, 10, 19, 20]). All of these results concern characterising the combinatorial nature
of the ‘generic’ behaviour of frameworks. This article extends this to frameworks with two
points lying in the same location. The difficulty that already arises in this context shows
how necessary the genericity assumption in those papers really was. Frameworks with
coincident points have been considered in the Euclidean context [12, 14] and applied to
global rigidity there [4], as well as for frameworks on surfaces [17].
Beyond the natural extension towards non-generic frameworks (and thus nearer to be-
ing of potential use in applications), we are motivated by the study of global rigidity in
non-Euclidean normed planes. The first and third author recently instigated research in this
direction [10] proving global rigidity for an infinite class of graphs in non-Euclidean ana-
lytic normed planes. In this paper we use our analysis of frameworks with two coincident
points to improve this result by creating a substantially richer class of globally rigid graphs.
We now give a short outline of what follows. After introducing the necessary back-
ground on the theory of rigid frameworks in normed planes, coincident-point frameworks
and the relevant notion of graph sparsity, in Section 2, the majority of the paper is con-
tained in Section 3. Here we provide a detailed geometric analysis of the effect of certain
graph operations on the rigidity of a coincident-point framework in a normed plane. In
Section 4 we combine these geometric results with combinatorial results of [17] to es-
tablish a purely combinatorial characterisation of independence in the ‘coincident-point
normed plane rigidity matroid’ and we deduce from this a delete-contract characterisation
of coincident-point rigidity in any strictly convex non-Euclidean normed plane. By delete-
contract characterisation we mean that we characterise the coincident-point rigidity of a
graph G in terms of the rigidity of two graphs related to G; the graphs obtained from G
by deleting the edge between the coincident vertices and the graph obtained by contracting
the two coincident vertices. In Section 5 we provide our other main results; these concern
global rigidity. We deduce from our delete-contract characterisation that another graph
operation preserves global rigidity, and we use this result alongside the results of [10] to
establish global rigidity in the special case of non-Euclidean analytic normed planes for a
rich family of graphs.
We conclude the introduction with a brief comparison with the more familiar Euclidean
case to give context for the reader. Both our characterisation of independence in the
coincident-point normed plane rigidity matroid and our delete-contract characterisation are
precise analogues of results obtained by Fekete, Jordán and Kaszanitzky for the Euclidean
case [12]. Furthermore, in the Euclidean case generic global rigidity in the plane is com-
pletely characterised [16]. Our results provide a key step towards establishing an analogue
of that result in non-Euclidean analytic normed planes. It is worth noting though that the
S. Dewar et al.: Coincident-point rigidity in normed planes 3
results of [12] came later than the Euclidean plane characterisation and, to our knowledge,
have not been used to provide an alternative proof of the global rigidity characterisation in
the Euclidean plane. The non-Euclidean normed plane case requires both subtly different
combinatorics and geometry which motivated our deployment of this technique. We would
expect that our application to global rigidity through ‘generalised vertex splitting’ (defined
in Section 5) could be adapted to the Euclidean case.
2 Rigidity and uv-coincident frameworks in normed spaces
2.1 Rigidity in normed spaces
Let X be a real finite-dimensional normed space with norm ∥ · ∥. We define a support
functional of z ∈ X to be a linear functional f : X → R such that f(z) = ∥z∥2 and
sup∥x∥=1 f(x) = ∥z∥. It follows from the Hahn-Banach theorem that every point has a
support functional and every linear functional of X is the support functional of a point in
X . A non-zero point in X is said to be smooth if it has exactly one support functional,
and we shall denote the unique support functional of a smooth point z by φz . We say X is
smooth if every non-zero point in X is smooth, and strictly convex if every linear functional
of X is the support functional of at most one, and hence exactly one, point in X .1 We note
that for normed planes (2-dimensional normed spaces), strict convexity is equivalent to the
property that any two linearly independent smooth points have linearly independent support
functionals.
Now let (G, p) be a framework in X; that is the combination of a graph G = (V,E)
and a map p : V → X (called a placement of G). A finite flex of (G, p) is a continuous path
α : [0, 1] → XV where α(0) = p and ∥αx(t)−αy(t)∥ = ∥px − py∥ for each edge xy ∈ E
and every t ∈ [0, 1]. If every framework (G,α(t)) is congruent to (G, p), i.e. there exists
an isometry ft : X → X so that αx(t) = ft(px) for every x ∈ V , then we say α is trivial.
We now define (G, p) to be (continuously) rigid if every finite flex of (G, p) is trivial.
Since determining whether a framework is rigid is computationally challenging [1],
we follow the literature and linearise the problem. First, let (G, p) be a well-positioned
framework, i.e. the point px − py is smooth for each edge xy ∈ E. An infinitesimal
flex of (G, p) is a map u : V → X where φpx−py (ux − uy) = 0 for all xy ∈ E. An
infinitesimal flex is trivial if there exists a linear map T : X → X and a point z0 ∈ X
so that ux = T (px) + z0 for every vertex x ∈ V , and the map T is tangent to the linear
isometry group of X at the identity map. Importantly, when X has finitely many linear
isometries – for example, when X is a non-Euclidean normed plane [25, page 83] – the
only trivial infinitesimal flexes are those that stem from translations, i.e., infinitesimal flexes
u = (ux)x∈V where there exists z ∈ X such that ux = z for all x ∈ V . We now say that
a well-positioned framework (G, p) is infinitesimally rigid if every infinitesimal flex of
(G, p) is trivial.
For a d-dimensional normed space X , a well-positioned framework (G, p) in X , and a
fixed basis b1, . . . , bd of X , we can define the rigidity matrix to be the |E| × d|V | matrix
1Here we have opted to use a more relevant – but still equivalent – definition for strict convexity. The more
conventional definition for the property is as follows: a normed space is said to be strictly convex if ∥tx + (1 −
t)y∥ < 1 for all x, y ∈ X with ∥x∥ = ∥y∥ = 1 and each 0 < t < 1. To see the equivalence, note that if
sup∥z∥=1 f(z) = 1 and ∥x∥ = ∥y∥ = 1, then f(x) = f(y) = 1 if and only if ∥tx+(1− t)y∥ = 1 for all 0 <
t < 1: this latter fact stems from the inequality tf(x)+(1−t)f(y) ≤ f(tx+(1−t)y) ≤ ∥tx+(1−t)y∥ ≤ 1.
4 Ars Math. Contemp. 24 (2024) #P1.10
R(G, p), where for every e ∈ E, x ∈ V and i ∈ {1, . . . , d} we have
R(G, p)e,(x,i) =
{
φpx−py (bi) if e = xy,
0 otherwise.
The choice of basis used to define R(G, p) can be arbitrary as we are only interested in
the sets of linearly independent rows of the matrix. We say a well-positioned framework
is independent if rankR(G, p) = |E|, minimally (infinitesimally) rigid if it is both in-
dependent and infinitesimally rigid, and regular if rankR(G, p) ≥ rankR(G, q) for all
other well-positioned frameworks (G, q). It is immediate that all independent and/or in-
finitesimally rigid frameworks are regular. Given k is the dimension of the linear space of
trivial infinitesimal flexes of (G, p), it can be shown that so long as the affine span of the
set {px : x ∈ V } is X , the framework (G, p) will be infinitesimally rigid if and only if
rankR(G, p) = d|V | − k; see [6, Proposition 3.13]. Consequently any well-positioned
framework where the affine span of the set {px : x ∈ V } is X , is minimally rigid if and
only if |E| = rankR(G, p) = d|V | − k.
We can link infinitesimal rigidity to rigidity with the following result.
Theorem 2.1. Let (G, p) be a well-positioned framework in a normed space X .
(i) [7, Theorem 3.7] If (G, p) is infinitesimally rigid, then it is rigid.
(ii) [6, Theorem 1.1 & Lemma 4.4] If (G, p) is regular and rigid, and the set of smooth
points in X is open, then (G, p) is infinitesimally rigid.
We shall make use of the following perturbation result throughout the paper. It will be
convenient to refer to properties of placements rather than frameworks. To this end we say
that a placement p of G has property P if the framework (G, p) has property P .
Lemma 2.2 ([6, Lemmas 4.1 and 4.4]). For any graph G and any normed space X , the
set of well-positioned placements of G in X is a conull (i.e. the complement of a set with
Lebesgue measure zero) subset of XV , and the set of regular placements of G in X is a
non-empty open subset of the set of well-positioned placements.
We say that a graph is rigid (respectively, independent, minimally rigid) if it has an
infinitesimally rigid (respectively, independent, minimally rigid) placement.
Whether a graph G = (V,E) is rigid/independent in a normed plane can be determined
by simple sparsity counting conditions. For ∅ ̸= U ⊆ V , iG(U) will denote the number
of edges in the subgraph, G[U ], of G induced by U . For non-negative integers k, ℓ, we
say G is (k, ℓ)-sparse if iG(U) ≤ k|U | − ℓ for every ∅ ̸= U ⊆ V with |U | ≥ k; if G is
(k, ℓ)-sparse and |E| = k|V | − ℓ, then we say G is (k, ℓ)-tight.
Theorem 2.3 ([5]). A graph G is minimally rigid in a non-Euclidean normed plane X if
and only if G is (2, 2)-tight.
For a family S = {S1, S2, . . . , Sk} of subsets Si ⊆ V , 1 ≤ i ≤ k, we say that S is
a cover of F ⊆ E if F ⊆ {xy : {x, y} ⊆ Si for some 1 ≤ i ≤ k}. We can combine
Theorem 2.3 with [17, Section 3.1] (which simply applies a classical result of Edmonds
[11] on matroids induced by submodular functions) to obtain the following result.
S. Dewar et al.: Coincident-point rigidity in normed planes 5
Corollary 2.4. Let (G, p) be a well-positioned framework in a non-Euclidean normed
plane X . Let S be the set of all covers X := {X1, . . . , Xk}. Given s : N → {0, 1} is the
map with s(x) = 1 if x = 2 and s(x) = 0 otherwise, we have
rankR(G, p) ≤ min
X∈S
k∑
i=1
(2|Xi| − 2− s(|Xi|)) ,
with equality if and only if (G, p) is regular. Moreover it suffices to minimise over all covers
Y := {Y1, . . . , Yk} of the edge set E where |Yi| ≥ 2 for each i and |Yi ∩ Yj | ≤ 1 for all
i ̸= j, with equality only if min{|Yi|, |Yj |} = 2.
2.2 uv-coincident rigidity and uv-sparse graphs
Let G = (V,E) be a graph with vertices u, v ∈ V , and let X be a normed space. A
framework (G, p) in X is uv-coincident if pu = pv; if the framework (G− uv, p) is well-
positioned, then we say that (G, p) is a well-positioned uv-coincident framework. Since
pu = pv , we consider G − uv so as to maintain smoothness of the support functionals
associated with the framework; otherwise, no uv-coincident framework with uv as an edge
would be well-positioned.
A well-positioned uv-coincident framework (G, p) is infinitesimally rigid if (G−uv, p)
is infinitesimally rigid in X . Given the linear space XV /uv := {q ∈ XV : qu = qv},
we say that a well-positioned uv-coincident framework (G, p) is regular if rankR(G −
uv, p) ≥ rankR(G − uv, q) for all q ∈ XV /uv, and independent if uv /∈ E and (G, p)
is independent in X . A well-positioned uv-coincident framework (G, p) is minimally (in-
finitesimally) rigid if it is both infinitesimally rigid and independent. We say a graph G is
uv-rigid (respectively, uv-independent, minimally uv-rigid) if there exists a uv-coincident
framework (G, p) that is infinitesimally rigid (respectively, independent, minimally rigid).
By applying the same methods used to prove Lemma 2.2, we can obtain the natural
analogue for uv-coincident frameworks. The two main observations for proving the result
are: (i) the set of smooth points of a normed space form a conull subset (i.e. the complement
of a set with Lebesgue measure zero) of X and (ii) the map p 7→ R(G − uv, p) is lower
semi-continuous.
Lemma 2.5. For any graph G and any normed space X , the set of well-positioned uv-
coincident placements of G in X is a conull subset of XV /uv, and the set of regular uv-
coincident placements of G in X is a non-empty open subset of the set of well-positioned
uv-coincident placements.
As will be shown in Section 4, uv-rigidity in non-Euclidean normed planes is closely
related to the following sparsity property of graphs.
Let G = (V,E) be a graph and let u, v be two distinct vertices of G. Let X =
{X1, X2, ..., Xk} be a family with Xi ⊆ V , 1 ≤ i ≤ k. We say that X is uv-compatible if
u, v ∈ Xi and |Xi| ≥ 3 hold for all 1 ≤ i ≤ k. We define the value of non-empty subsets
of V and of uv-compatible families, denoted val(·), as follows. For ∅ ≠ U ⊆ V , we let
val(U) = 2|U | − tU ,
where tU = 4 if U = {u, v}, tU = 3 if U ̸= {u, v} and |U | ∈ {2, 3}, and tU = 2
6 Ars Math. Contemp. 24 (2024) #P1.10
otherwise. For a uv-compatible family X = {X1, X2, . . . , Xk} we let
val(X ) =
(
k∑
i=1
val(Xi)
)
− 2(k − 1) = 2 +
k∑
i=1
(2|Xi| − tXi − 2).
Note that if X = {U} is a uv-compatible family containing only one set then the two
definitions agree, i.e. val(X ) = val(U) holds.
We say that G is uv-sparse if for all U ⊆ V with |U | ≥ 2 we have iG(U) ≤ val(U)
and for all uv-compatible families X we have iG(X ) :=
∣∣∣⋃ki=1 E(G[Xi])∣∣∣ ≤ val(X ). A
graph G is uv-tight if it is uv-sparse and |E| = 2|V | − 2. Note that if G is uv-sparse then
uv /∈ E. It was shown in [17] that if G = (V,E) is a graph and u, v ∈ V are distinct
vertices of G then I = {F : F ⊆ E and (V, F ) is uv-sparse} is the family of independent
sets of a matroid Muv on E.
It is straightforward to construct (2, 2)-sparse graphs which are not uv-sparse. Perhaps
the simplest way is to notice that the complete bipartite graph K2,3, with the part of size
two comprising of u and v, is clearly (2, 2)-sparse but fails to be uv-sparse. To see this
let v1, v2, v3 be the vertices in the part of size three and consider the uv-compatible family
X = {X1, X2, X3} where X1 = {u, v, v1}, X2 = {u, v, v2} and X3 = {u, v, v3}. Then
iG(X ) = 2+2+2 = 6 and val(X ) = (2 · 3− 3)+ (2 · 3− 3)+ (2 · 3− 3)− 2(3− 1) = 5.
3 Recursive operations
Let G = (V,E) be a graph. The 0-extension operation (on a pair of distinct vertices
a, b ∈ V ) adds a new vertex z and two edges za, zb to G. The 1-extension operation
(on edge ab ∈ E and vertex c ∈ V \ {a, b}) deletes the edge ab, adds a new vertex z
and edges za, zb, zc. The vertex-to-H move adds a copy of a (2, 2)-tight graph H with
V (H) ∩ V = {w}, along with an arbitrary replacement of each edge xw by an edge of
the form xy with y ∈ V (H). A vertex-to-4-cycle move takes a vertex w with neighbours
v1, v2, . . . , vk for any k ≥ 2, splits w into two new vertices w,w′ with w′ /∈ V , adds edges
wv1, w
′v1, wv2, w
′v2 and then arbitrarily replaces edges xw with edges of the form xy
where x ∈ {v3, . . . , vk} and y ∈ {w,w′}. All (2, 2)-tight graphs can be constructed from
a single vertex by a sequence of 0- and 1-extensions, vertex-to-4-cycle and vertex-to-K4
operations; see [22, Theorem 3.1] for more details. The operations we use are illustrated in
Figures 1 and 2.
Figure 1: 0-extension and 1-extension.
S. Dewar et al.: Coincident-point rigidity in normed planes 7
Figure 2: The vertex-to-H (with H being the complete graph on 4 vertices) and vertex-to-
4-cycle operations.
We shall need the following specialized versions. First, suppose that |V ∩ {u, v}| =
1. The 0-extension that adds u (respectively, 0-extension that adds v) operation is a 0-
extension where z = u and v ∈ V \ {a, b} (respectively, with z = v and u ∈ V \ {a, b}).
The vertex-to-4-cycle move that adds u (respectively, vertex-to-4-cycle move that adds v)
is a vertex-to-4-cycle move where w = v and {w,w′} = {u, v} (respectively, w = u and
{w,w′} = {u, v}). The vertex-to-H move that adds u (respectively, vertex-to-H move that
adds v) is a vertex-to-H move where w = v and u ∈ V (H) \ V (respectively, w = u and
v ∈ V (H) \ V ), and the graph H is uv-tight.
Now suppose u, v ∈ V are two distinct vertices. The uv-0-extension operation is a 0-
extension on a pair a, b with {a, b} ≠ {u, v}. The uv-1-extension operation is a 1-extension
on some edge ab and vertex c for which {u, v} is not a subset of {a, b, c}. The uv-vertex-
to-4-cycle and uv-vertex-to-H moves are simply any vertex-to-4-cycle and vertex-to-H
moves applied to a graph containing both u and v.
We can immediately obtain the following result using the proof technique of [5, Lem-
mas 5.1 and 5.2]. In particular, since the uv-0- and uv-1-extensions are local operations
that relate to at most one of u and v, their coincidence does not have any effect on the
proofs presented in [5].
Lemma 3.1. Let G be a graph that contains both u and v, and let G′ be formed from G by
either a uv-0-extension or a uv-1-extension. If G is uv-independent in a normed plane X ,
then G′ is uv-independent in X .
The next lemma shows 0-extensions that add either u or v preserve independence. It
should be noted that our proof technique requires strict convexity.
Lemma 3.2. Let G = (V,E) be a graph that contains u but not v, and let X be a strictly
convex normed plane. Suppose G′ is formed from G by a 0-extension that adds v. Then G′
is uv-independent if and only if G is independent.
Proof. We note that as G′ contains G as a subgraph, if G′ is uv-independent then G will
be independent. Suppose there is an independent placement p of G in X . By applying
translations, we may suppose that pu = 0. Let v1, v2 be the two neighbours of v in G′.
We may also assume that pv1 and pv2 are linearly independent and smooth; indeed if this
8 Ars Math. Contemp. 24 (2024) #P1.10
was not true, we could apply Lemma 2.2 to (G, p) to find a placement of G where it is
true. Define p′ to be the well-positioned placement of G′ with p′x = px for all x ∈ V and
p′v = pu. From our choice of placement of G
′, we see that
R(G′, p′) =
 R(G, p) 0|E|×2A −φpv1
B −φpv2

for some 1×2|V | matrices A and B. Hence (G′, p′) is independent if and only if φpv1 , φpv2
are linearly independent. Since pv1 , pv2 are linearly independent and X is strictly convex,
the pair φpv1 , φpv2 are linearly independent as required.
For the vertex-to-4-cycle move we will use the technique of [17, Lemma 11] to show
that a vertex-to-4-cycle move which creates two coincident vertices preserves indepen-
dence. Similarly to the previous result, we will require that the normed plane in question is
strictly convex.
Lemma 3.3. Let G = (V,E) and G′ = (V ′, E′) be graphs and let X be a strictly convex
normed plane.
(i) If G is independent in X and G′ is formed from G by a vertex-to-4-cycle move that
adds either u or v, then G′ is uv-independent in X .
(ii) If G is uv-independent in X and G′ is formed from G by a uv-vertex-to-4-cycle
move, then G′ is uv-independent in X .
Proof. Suppose that G is uv-independent (respectively, independent). Using Lemma 2.5
(respectively, Lemma 2.2), choose a uv-independent (respectively, independent) placement
p of G in X so that pw, pv1 , pv2 are not collinear. By applying translations to p, we shall
assume that pw = 0. Now define p′ to be the placement of G′ with p′x = px for all x ∈ V
and p′w′ = pw. The pair (G
′, p′) form a well-positioned uv-coincident framework due to
our choice of p′. Since X is strictly convex, the pair φpv1 , φpv2 are linearly independent.
Define G′′ to be the graph formed from G′ by replacing each edge w′vi for 3 ≤ i ≤ k with
the edge wvi. Then
R(G′′, p′) =
 R(G, p) 0|E|×2A φp′
w′−p
′
v1
B φp′
w′−p
′
v2
 =
 R(G, p) 0|E|×2A −φpv1
B −φpv2
 ,
for some 1 × 2|V | matrices A and B. Since pv1 , pv2 are linearly independent and X is
strictly convex, the pair φpv1 , φpv2 are linearly independent. Hence R(G
′′, p′) has linearly
independent rows. To prove that G′ is uv-independent in X we will describe a series of
rank-preserving row operations that will form R(G′, p′) from R(G′′, p′).
As φpv1 and φpv2 are linearly independent, there exist for each 3 ≤ i ≤ k a unique pair
of values αi and βi such that
αiφpv1 + βiφpv2 = φpvi = φp′vi−p
′
z
,
where z ∈ {w,w′} is chosen so that viz ∈ E′. For 1 ≤ i ≤ k, let (wvi) denote the row of
R(G′′, p′) corresponding to the edge wvi, and similarly let (w′v1) and (w′v2) denote the
S. Dewar et al.: Coincident-point rigidity in normed planes 9
rows of R(G′′, p′) corresponding to edges w′v1 and w′v2 respectively. For vi ∈ NG′(w′),
let [w′vi] denote the row of R(G′, p′) corresponding to the edge w′vi. Now, for all vi ∈
NG′(w
′)\{v1, v2}, we have
[w′vi] = (wvi)− αi(wv1)− βi(wv2) + αi(w′v1) + βi(w′v2).
These row operations, when applied R(G′′, p′), preserve linear independence and form the
matrix R(G′, p′). Therefore the rows of R(G′, p′) are linearly independent.
We now prove that vertex-to-H operations that add either u or v and uv-vertex-to-H
operations will preserve uv-independence.
Lemma 3.4. Let G = (V,E) and G′ = (V ′, E′) be graphs and let X be any non-
Euclidean normed plane.
(i) Suppose G is independent in X and G′ is formed from G by a vertex-to-H move that
adds either u or v. If H is minimally uv-rigid in X , then G′ is uv-independent in X .
(ii) Suppose G is uv-independent in X and G′ is formed from G by a uv-vertex-to-H
move. If H is minimally rigid in X , then G′ is uv-independent in X .
Proof. If (i) holds, let (H, q) be a minimally rigid uv-coincident framework in X and
(G, p) be an independent framework in X , while if (ii) holds, let (H, q) be a minimally
rigid framework in X and (G, p) be an independent uv-coincident framework in X . By
applying translations we may assume qw = pw = 0. For any matrix A with columns
corresponding to a vertex subset of V ∪V (H), define Aw to be the matrix where we delete
all columns corresponding to the vertex w. Given the fixed basis b1, b2 ∈ X used to define
our rigidity matrices in X , we define the matrix
M :=
[
R(H, q)w 0|E(H)|×(2|V |−2)
A R(G, p)w
]
where A is the |E| × (2|V (H)| − 2) matrix with entries
Ae,(y,i) =
{
φpy−pw(bi) if e = xw,
0 otherwise.
By our choices of p and q, the matrix M has linearly independent rows.
For each n ∈ N, choose a well-positioned uv-coincident framework (G′, pn) where
pnx = qx/n for each x ∈ V (H) and ∥pnx − px∥ < 1/n for each x ∈ V (this framework can
be seen to exist from Lemma 2.5). Define Mn to be the matrix formed from multiplying
each row of R(G′, pn)w corresponding to an edge of H by n. As the map x 7→ φx is
continuous on the set of smooth points of X (see [24, Theorem 25.5]), the sequence of
matrices (Mn)n∈N will converge to M . Hence for sufficiently large N ∈ N, the matrix
Mn0 (and hence R(G
′, pn0)w) will have linearly independent rows. By setting p′ = pn0 ,
we obtain our desired independent uv-coincident framework (G′, p′).
10 Ars Math. Contemp. 24 (2024) #P1.10
4 Characterising coincident-point independence
With the geometric results of the previous section in hand, we can use the combinatorics
of [17] to prove the difficult sufficiency direction of our main result on coincident frame-
works. We begin with the following result which can be extracted from the proof of [17,
Theorem 4].
Proposition 4.1 ([17]). Any uv-tight graph on at least five vertices can be constructed
from either a (2, 2)-tight graph with at least four vertices that contains exactly one of u
and v, or from the graph consisting of two copies of K4 intersecting in a single vertex
x /∈ {u, v} where u and v are in different copies of K4 (see Figure 3), by a sequence of
0-extensions that add u or v, vertex-to-4-cycle and vertex-to-H moves that add u or v,
uv-0- and uv-1-extensions, and uv-vertex-to-4-cycle and uv-vertex-to-H moves.
Sketch of proof. Since the proof of [17, Theorem 4] is long and technical we provide a
sketch of the proposition to orient the interested reader into how it can be extracted from
that theorem. It is easy to see that every graph generated as described in the statement
is uv-tight. For the converse, firstly [17, Theorem 4] is stated for independence in Muv ,
i.e. for uv-sparse graphs, but we can extend to a base E of Muv which induces a graph
G = (V,E) that, since |V | ≥ 5, necessarily has 2|V | − 2 edges and hence is uv-tight.
Suppose G has a vertex, w, of degree 2. If w ∈ {u, v} then G − w is (2, 2)-tight. If
w /∈ {u, v} then an easy argument shows that G− w is uv-tight.
Therefore we may assume that the minimum degree is exactly 3, however it is much
harder to reduce degree 3 vertices. [17, Theorem 4] deals firstly with three straightforward
special cases. Firstly, if there is a 4-cycle in G containing u and v then uv is not an edge of
this 4-cycle and we see that G is obtained from a (2, 2)-tight graph by a vertex-to-4-cycle
operation that split u into u and v. Secondly, if G contains a uv-tight subgraph H such that
V (H) ⊊ V then we may assume H is a maximal such subgraph (that is there is no vertex
in V \ V (H) with more than one neighbour in V (H)). Then G/H (the graph obtained
from G by contracting all vertices of H to a single vertex) is (2, 2)-tight and G is obtained
from G/H by a vertex-to-H move that expands u into a subgraph H that contains u and
v. Thirdly, if G contains a degree 3 vertex contained in a subgraph of G isomorphic to K4
and there is a vertex x ∈ V \ V (H) such that |V (H) ∩ N(x)| = 2 (and since we may
assume the second special case does not occur {u, v} ̸⊂ V (H) ∪ {x}), then we may apply
a uv-vertex-to-H move to a uv-tight graph G/(H ∪ {x}) to obtain G.
The proof is then completed by applying the arguments in Cases 5 and 6 of [17, The-
orem 4], which use the fact we do not have the special structures we just dealt with, to
analyse all possibilities for reducing a vertex of degree 3. Note that it is still not true that
1-extensions and uv-1-extensions suffice, however it is true that using precisely the opera-
tions listed in the proposition is sufficient.
We will also require the following lemmas.
Lemma 4.2. Let G = (V,E) be a graph with at most 4 vertices that contains both u and
v, and let X be a strictly convex non-Euclidean normed plane. Then G is uv-sparse if and
only if it is uv-independent in X .
Proof. The only graphs on 4 or fewer vertices that are not uv-sparse are those which con-
tain the edge uv, and if G contains the edge uv then it is not uv-independent. Suppose
uv /∈ E. We note that G must be a subgraph of K4 − uv, so it is sufficient to consider the
S. Dewar et al.: Coincident-point rigidity in normed planes 11
vu
x
Figure 3: A uv-tight graph that is one of the base graphs of the construction described in
Proposition 4.1.
case G = K4 − uv. As G can be formed from G − u by a 0-extension that adds u, G is
uv-independent by Theorem 2.3 and Lemma 3.2.
Lemma 4.3. Let G = (V,E) be the graph consisting of two copies of K4 intersecting in a
single vertex x /∈ {u, v}, where u and v are in different copies of K4 (see Figure 3). Then
G is minimally uv-rigid in any non-Euclidean normed plane X .
Proof. Let Vu = {x, u, au, bu} and Vv = {x, v, av, bv} be the vertex sets of the two copies
of K4 in G. As can be seen in Figure 3, Vu ∩ Vv = {x}. By Theorem 2.3, there exists
a placement pu : Vu → X so that the framework (KVu , pu), where KVu is the complete
graph with vertex set Vu, is minimally rigid in X . Define the placement p : V → X by
setting pav = p
u
au , pbv = p
u
bu
, pv = puu, and py = p
u
y for all y ∈ Vu. We now note that
(G, p) is a minimally rigid uv-coincident framework; this follows from the fact that joining
two minimally rigid frameworks in a non-Euclidean normed plane produces a minimally
rigid framework, since the trivial infinitesimal flexes of a non-Euclidean normed plane
correspond only to translations. Hence G is minimally uv-rigid as required.
Theorem 4.4. A graph is uv-independent in a strictly convex non-Euclidean normed plane
X if and only if it is uv-sparse.
Proof. First suppose G is uv-independent in X . Let G/uv denote the graph obtained from
G by contracting the vertex pair u, v into a new vertex which we denote as z2. Let (G, p) be
a regular (and hence independent) uv-coincident framework in X . We obtain a framework
(G/uv, puv) in X by putting puvz = pu = pv and p
uv
x = px for all x ∈ V \ {u, v}. For any
U ⊆ V , the (possibly uv-coincident) induced subframework (G[U ], p|U ) is independent.
Hence, if {u, v} ̸⊆ U , then iG(U) ≤ val(U) by Theorem 2.3. Since the case when
U = {u, v} is trivial, it now remains to show that iG(X ) ≤ val(X ) for all uv-compatible
families X in G. (Note that the case when U ⊆ V and {u, v} ⊆ U will be included by
taking X = {U}.)
Let X = {X1, . . . , Xk} be a uv-compatible family and consider the subgraph H =
(U,F ) of G, where U =
⋃k
i=1 Xi and F =
⋃k
i=1 E(G[Xi]). By contracting the vertex
pair u, v in H , we obtain the graph H/uv. Define q to be the restriction of p to the vertex
set U and quv to be the restriction of puv to the vertex set U − {u, v} + z. We have
Xuv = {X1/uv, . . . ,Xk/uv} is a cover of E(H/uv) where Xi/uv denotes the set that we
2For us, a contraction will always be the more general vertex-contraction (which does not require u and v be
adjacent) not the stricter edge-contraction (which does require u and v be adjacent).
12 Ars Math. Contemp. 24 (2024) #P1.10
get from Xi by identifying u and v. By Corollary 2.4, we have
rankR(H/uv, quv) ≤
k∑
i=1
(2|Xi/uv| − 2− s(|Xi/uv|)
=
k∑
i=1
(2|Xi| − 2− tXi)
= val(X )− 2.
Every vector µuv in the kernel of R(H/uv, quv) determines a unique vector µ in the kernel
of R(H, q) with µu = µv = µuvz and µx = µ
uv
x for all for all x ∈ U \ {u, v}. Hence
dimkerR(H, q) ≥ dimkerR(H/uv, quv). The rigidity matrix R(H, q) has linearly inde-
pendent rows since R(G, p) has linearly independent rows, hence we have
iG(X ) = rankR(H, q) ≤ rankR(H/uv, quv) + 2 ≤ val(X ).
Thus G is uv-sparse.
We prove the sufficiency by induction on |V |. Suppose that G is uv-sparse. If |V | ≤ 4,
then G is uv-independent in X by Lemma 4.2. So we may suppose that |V | ≥ 5. By
adding additional edges, if necessary, we may assume G is uv-tight3. By Proposition 4.1,
G can be constructed from either a (2, 2)-tight graph containing exactly one of u and v, or
the graph pictured in Figure 3, by the operations defined in Section 3. Furthermore, as X
is strictly convex, the corresponding geometric operations preserve minimal rigidity in X
(see Section 3). The result now follows from Theorem 2.3 (i.e., every (2, 2)-tight graph is
independent in X) and Lemma 4.3.
We next use this result to prove the following delete-contract characterisation of uv-
rigidity.
Theorem 4.5. Let G be a graph with distinct vertices u, v, and let X be a strictly convex
non-Euclidean normed plane. Then G is uv-rigid in X if and only if G−uv and G/uv are
both rigid in X .
Proof. Suppose that G is uv-rigid. It is immediate from the definition that G−uv must be
rigid. Choose a regular uv-coincident placement p of G, and define puv to be the placement
of G/uv where puvx = px for all x ∈ V − {u, v} and (given that z is the vertex obtained
from u and v during the contraction) puvz = pu = pv . Given an infinitesimal flex µ
uv
of (G/uv, puv), we can form an infinitesimal flex µ of (G, p) by setting µx = µuvx for
all x ∈ V − {u, v} and µu = µv = µuvz . Since (G, p) is infinitesimally rigid as a uv-
coincident framework, we must have that µ = (λ)x∈V (and hence µuv = (λ)x∈V (G/uv))
for some vector λ ∈ X . Thus (G/uv, puv) is infinitesimally rigid and G/uv is rigid.
The converse follows from Theorem 4.4 as in the proof of [17, Theorem 1].
We conjecture that the last two results apply in arbitrary non-Euclidean normed planes.
Conjecture 4.6. Let G = (V,E) be a graph and let u, v ∈ V be distinct vertices. Then G
is uv-independent in a non-Euclidean normed plane X if and only if G is uv-sparse.
3Recall that uv-sparse graphs are the independent sets of a matroid and, when |V | ≥ 5, the bases of this
matroid have rank 2|V | − 2.
S. Dewar et al.: Coincident-point rigidity in normed planes 13
Indeed extending the proof of Theorem 4.4 to the non-convex case requires only im-
provements to Lemmas 3.2 and 3.3. For the first of these, the issue is that 0-extensions
that add v require us to precisely place v on top of the placement of u. However in the
not strictly convex case, the proof of [5, Lemmas 5.1] requires one to choose the position
of v carefully so that the support functionals of the edges incident to v guarantee linear
independence. For the latter case, both the vertex-to-4-cycle move that adds v and the
uv-vertex-to-4-cycle move have similar complications that would need to be resolved.
5 Global rigidity in analytic normed planes
A framework (G, p) in a normed space X is said to be globally rigid if every other frame-
work (G, q) in X with ∥pv − pw∥ = ∥qv − qw∥ for every edge vw ∈ E is congruent to
(G, p). A graph is then said to be globally rigid in X if the set{
p ∈ XV : (G, p) is globally rigid
}
has a non-empty interior as a subset of the linear space XV with the product topology
inherited from X . It can be quickly seen that any globally rigid framework/graph will also
be rigid.
Although much is known about global rigidity in Euclidean spaces, very little is known
about the property for non-Euclidean normed spaces. The results that are known are only
for analytic normed spaces, i.e., normed spaces where the norm restricted to the non-zero
points is a real analytic function. As well as being strictly convex ([10, Lemma 3.1]),
analytic normed spaces have many useful properties, including the following.
Lemma 5.1. Let G be a graph with distinct vertices u, v and let X be a non-Euclidean
analytic normed space.
(i) The set of all p ∈ XV where (G, p) is a regular framework is an open conull subset
of XV .
(ii) The set of all p ∈ XV /uv where (G, p) is a regular uv-coincident framework is an
open conull subset of XV /uv.
Proof. If dimX = 1 then the result follows immediately from noticing that all well-
positioned frameworks and uv-coincident frameworks are regular. Suppose dimX ≥ 2.
It was shown in [10, Proposition 3.2] that the set of well-positioned but non-regular place-
ments of G are exactly the zero set of a non-constant analytic function defined on the con-
nected open conull set of well-positioned placements. This gives (i). For (ii) we can use
the same technique to show that the set of well-positioned but non-regular uv-coincident
placements of G are exactly the zero set of a non-constant analytic function defined on the
connected open conull set of well-positioned uv-coincident placements. The result now
holds as the zero set of a non-constant analytic function with connected domain is always
a closed null subset (see [10, Proposition 2.3]).
Importantly, we can define a large class of globally rigid graphs in any non-Euclidean
analytic normed plane.
Proposition 5.2 ([10]). Let X be a non-Euclidean analytic normed plane. Then the graphs
K5−e and H , depicted in Figure 4, are globally rigid in X . Moreover any graph obtained
from either of these by a sequence of degree 3 vertex additions (i.e., add a vertex and join
it to three other vertices) and edge additions is globally rigid.
14 Ars Math. Contemp. 24 (2024) #P1.10
Figure 4: The graphs K−5 (left) and H (right).
We next increase this class of graphs with the following construction operation intro-
duced in [18]. A generalised vertex split, is defined as follows. Choose z ∈ V and a parti-
tion Nu, Nv of the neighbours of z. Next, delete z from G and add two new vertices u, v
joined to Nu, Nv , respectively. Finally add two new edges uv, uw for some w ∈ V \Nu.
See Figure 5 for an illustration of the operation.
z u v
w w
Figure 5: Generalised vertex split.
As the name suggests, this operation generalises the usual vertex splitting operation,
see [26], which is the special case when w is chosen to be a neighbour of v. Note also that
the special case when u has degree 3 (and v = z) is the well known 1-extension operation.
Previously it was not known whether the 1-extension operation or a suitably restricted
version of the vertex splitting operation preserves global rigidity in any non-Euclidean
normed plane X .
As an application of our main result we will deduce that global rigidity can, under cer-
tain conditions, be preserved for generalised vertex splits. We will first need the following
result which can be seen to follow from adapting the methods in [10, Section 3.2] to allow
frameworks with zero-length edges4.
Lemma 5.3. Let (G, p) be a uv-coincident framework in a smooth normed space X with
finitely many linear isometries. If (G, p) is globally rigid and infinitesimally rigid, then
there exists an open neighbourhood U ⊂ XV of p where for each q ∈ U the framework
(G, q) is globally rigid.
Theorem 5.4. Let G be a globally rigid graph in a non-Euclidean analytic normed plane
X . Let G′ be a generalised vertex split of G at the vertex z with new vertices u, v and
suppose that G′ − uv is rigid in X . Then G′ is globally rigid in X .
4Although it is a prerequisite in [10, Section 3.2] that the frameworks are well-positioned, the proof technique
only requires that the squared edge-length map is differentiable. Since the map x 7→ ∥x∥2 is always differentiable
at the point 0, we can refine the result so that it holds for frameworks with zero-length edges.
S. Dewar et al.: Coincident-point rigidity in normed planes 15
Proof. Since G′/uv = G is globally rigid in X it is also rigid in X by Theorem 2.1.
As G′ − uv is also rigid in X , Theorem 4.5 implies that G′ is uv-rigid in X . Hence by
Lemma 5.1, we may choose an infinitesimally and globally rigid framework (G, p) so that
if we define (G′, p′) to be the uv-coincident framework with p′x = px for all x ∈ V and
p′u = p
′
v = pz , then (G
′, p′) will be infinitesimally rigid also. Furthermore, (G′, p′) will
also be globally rigid as (G, p) is globally rigid. We can now use Lemma 5.3 to deduce that
(G′, q) is globally rigid in X for all q sufficiently close to p′. Hence G′ is globally rigid in
X also.
We can now improve upon Proposition 5.2. Here a graph G = (V,E) is redundantly
rigid in X if G− e is rigid in X for any edge e ∈ E.
Corollary 5.5. Let G be a graph obtained from K−5 or H by a sequence of generalised
vertex splits that preserve redundant rigidity, edge additions and degree at least 3 vertex
additions. Then G is globally rigid in any non-Euclidean analytic normed plane.
Proof. Follows immediately from Proposition 5.2 and Theorem 5.4.
Since minimally rigid graphs in X have 2|V | − 2 edges by Theorem 2.3, it is natural
to expect that if G = (V,E) is globally rigid then |E| ≥ 2|V | − 1. The graphs K−5 and
H both achieve equality, but the inequality is strict for every graph in the infinite family
obtained from these as in Proposition 5.2. To illustrate the power of Corollary 5.5 we note
that we now have infinitely many globally rigid graphs for which equality holds and that
this still holds if we restrict generalised vertex splitting to just one of vertex splitting or
1-extension. Two examples are depicted in Figure 6. The graph on the left is obtained from
H by a vertex split and the graph on the right is obtained from H by a 1-extension. Both
are globally rigid in X by Corollary 5.5.
Figure 6: Examples of globally rigid graphs.
6 Concluding remarks
1. Following submission of this article we were able to improve upon Corollary 5.5. Specif-
ically in [8], using the results of this article in a crucial way, we obtained a complete
combinatorial description of graphs that are globally rigid in any non-Euclidean analytic
normed plane. It turns out that we needed just 1 additional operation to those used in Corol-
lary 5.5: this operation deletes an edge xy and adds two new vertices z, w and 5 new edges
xz, xw, yz, yw, zw. In different language, the characterisation of [8] shows that a graph is
globally rigid in any non-Euclidean analytic normed plane if and only if it is 2-connected
and redundantly rigid (which means that it is still rigid after deleting any edge).
16 Ars Math. Contemp. 24 (2024) #P1.10
2. Theorem 4.4 and Theorem 4.5 provide a detailed combinatorial understanding of coin-
cident point rigidity for frameworks in strictly convex non-Euclidean normed planes. As
noted in the introduction, similar results exist for the Euclidean plane [12] and for frame-
works supported on a cylinder in R3 [17]. Given the applicability of coincident point
rigidity to analysing global rigidity (e.g. [4]) it would be interesting to develop analogues
of Theorem 4.4 and Theorem 4.5 in other natural settings in rigidity theory. It may also be
interesting to explore rigidity for frameworks with larger (or multiple) sets of coincident
points. This line of investigation has begun in the case of the Euclidean plane [15].
ORCID iDs
Sean Dewar https://orcid.org/0000-0003-2220-4576
John Hewetson https://orcid.org/0000-0001-9369-7895
Anthony Nixon https://orcid.org/0000-0003-0639-1295
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