on n vertices, Kn = (V, K(V)). An edge e∈K(V)\E is independent of E if the row
of R(Kn, p) corresponding to the edge e is independent of the rows corresponding to
the edges in E. If e is not independent of E, then it is induced by E.
The vertex set of E ⊂ K(V) is denoted by V(E), where v ∈ V(E) if and only
if {v, w} ∈ E for some vertex w ∈ V. When every vertex is contained in V(E) (i.e. V(E) = V),E may be referred to as a spanning edge set for Kn= (V, K(V)).
The graph with edge set K(V)\E may be denoted byKn\E.
Remark 2.6.6 (Lemma 2.5.1 [GSS93]). Let G = (V, E). It may be useful to note
that an edge {i, j} ∈K(V) being independent of the edge set E is equivalent to the
regular framework (G, p) having an infinitesimal motion p′ such that [p(i)−p(j)]·
[p′(i)−p′(j)]̸= 0.
Definition 2.6.7 ([GSS93]). An edge set E is rigid if the framework (G, p), where G= (V(E), E) and p∈Rdn is any regular configuration, is infinitesimally rigid.
Lemma 2.6.8 (Lemma 2.5.2 [GSS93]). Let V be a set of n vertices, where Kn =
(V, K(V)) denotes the complete graph on the vertex set V. Let p∈Rdn be a regular
configuration of V, and let E ⊆K(V) be a spanning edge set (i.e. V(E) =V). The
edge set E is rigid if and only if every edge f ∈K(V)\E is induced by the edge set E.
2.7
Combinatorial rigidity
From Theorem 2.4.9 we know that an infinitesimally rigid framework (G, p) has
rankR(G, p) = dn−d+12 . Definition 2.4.14 defines the regular configurations of
a framework to be those p ∈ Rdn for which rankR(G, p) is maximal in
Rdn, so we
2.7 Combinatorial rigidity 33 infinitesimally flexible for all q∈Rdn. If instead (G, p) is infinitesimally rigid for some
regular p∈Rdn, then (G, q) is infinitesimally rigid for all regular q∈
Rdn. This may
be formally stated as follows.
Lemma 2.7.1 (Corollary 2 [AR78]). If a framework (G, p) is infinitesimally rigid
for some configuration p ∈ Rdn, then (G, q) is infinitesimally rigid for all regular
configurations q ∈Rdn.
It may also be useful to note that by Theorem 2.4.17, we know that for frameworks (G, p) with a regular configuration p ∈ Rdn, infinitesimal rigidity and continuous
rigidity are equivalent. When p is regular, rather than consider the rigidity of specific
realisations of frameworks, we may consider rigidity as a property of the graph G.
Definition 2.7.2. A graph G is generically rigid in dimension d if there is a generic
configuration p∈Rdn such that the framework (G, p) is infinitesimally rigid. If G is
also independent, we may refer to Gas being d-isostatic.
We may now characterise frameworks inRd that are isostatic for all regular con-
figurations p ∈ Rdn, by characterising d-isostatic graphs. We may therefore apply
combinatorial and graph theoretic techniques while avoiding frameworks in singular positions.
From the rigidity matrix R(G, p) and Theorem 2.4.11, it seems useful to define
certain sparsity counts for graphs. For d-isostatic graphs, we require that |E| =
d|V| − d+12 , and we note that for an independent framework we will also have |E(V′)| ≤d|V′| −d+12 for any subgraph generated byV′ ⊂V with |V′| ≥d. We give
the following standard definition in general, noting that the counts in Theorem 2.4.11 are equivalent to the case where k =d and ℓ=d+12 .
Definition 2.7.3. A finite graphG is (k, ℓ)-sparse if, for all subgraphsG′ = (V′, E′)
with|V′| ≥k,|E′| ≤k· |V′| −ℓ. IfGalso satisfies|E|=k· |V| −ℓ, thenGis (k, ℓ)-tight.
2.7 Combinatorial rigidity 34 A (k, ℓ)-circuit is a graph G where the removal of any edge e ∈ E results in a
(k, ℓ)-tight graphG−e. This is equivalent to |E|=k|V| −ℓ+ 1, with |E′| ≤k· |V′| −ℓ
for all proper subgraphs G′ = (V′, E′) with|V′| ≥k.
IfG is a (k, ℓ)-sparse graph with a (k, ℓ)-tight subgraphG′, the subgraph G′ is a
(k, ℓ)-block.
Remark 2.7.4. Recall the definition of a matroid Section 2.6: they may be viewed either as a collection of independent subsets of the ground set E, or in terms of a rank
function on the subsets of E, where D⊆E is independent if and only ifr(D) =|D|.
The subsets of rows of the rigidity matrix of the complete graph onn vertices, R(G, p),
that are independent correspond straightforwardly to the independent sets of the infinitesimal rigidity matroid Md(G) (Definition 2.6.3).
The independent sets of the (k, ℓ)-sparsity matroid [LS08, Whi96] on an edge set E
are precisely all (k, ℓ)-sparse subgraphs. These matroids are characterised for integers k, ℓ such that 0≤ℓ ≤2k−1 by the (k, ℓ)-pebble game algorithms [LS08].
The minimal dependent sets of a matroid are referred to as the circuits of the
matroid, and the circuits in the (k, ℓ)-sparsity matroid correspond precisely to the
(k, ℓ)-circuits as defined above.
Remark 2.7.5. Following convention, we refer to graphs that are (2,3)-tight asLaman graphs, and (2,3)-sparse graphs as Laman-sparse graphs. We may also refer to (2,3)-
tight subgraphs within a graph as rigid blocks, and (2,3)-circuits asrigidity circuits (or circuits where the context is clear).
Remark 2.7.6. We note that some authors refer to graphs that are d,d+12 -tight
as satisfying the d-dimensional Maxwell-Laman conditions.
Theorem 2.7.7 ([Max64a, GSS93]). If a framework (G, p) that affinely spans Rd is d-isostatic, the graph Gis d,d+12 -tight.
2.8 Inductive constructions 35