The Rigidity of Graphs
Author(s): L. Asimow and B. Roth
Source: Transactions of the American Mathematical Society, Vol. 245 (Nov., 1978), pp. 279-289
Published by: American Mathematical Society
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TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 245, November 1978
TIHERIGIDrITY
OF GRAPHS
BY
L. ASIMOW AND B. ROTH
ABSTRACT. We regard a graph G as a set {1,...,v)
together with a
nonemptyset E of two-elementsubsetsof (1, . . ., v). Letp = (Pi, . . . &)
be an elementof RIvrepresentingv points in RI. Considerthe figureG(p)
in RI consistingof the line segments[pi,pj in RI for {i,j) E E. The figure
G(p) is said to be rigidin RI if everycontinuouspathin RI', beginningatp
and preservingthe edge lengths of G(p), terminatesat a point q E RIV
which is the image (Tp1,. . ., Tpv) of p under an isometry T of RW.
Otherwise,G(p) is flexiblein RW.Our main resultestablishesa formulafor
determiningwhetherG(p) is rigid in RI for almost all locationsp of the
vertices.Applicationsof the formulaare made to completegraphs,planar
graphs,convex polyhedrain R3,and otherrelatedmatters.
1. Introduction.Considera triangleor a squarein R2 for which the edges
are rods which arejoined but rotate freely at the vertices.The squareis said
to be flexible in R2 since the squarecan move continuouslyinto a family of
rhombi. However, the triangleis said to be rigid in R2 since the three rods
determinethe relativepositionsof the threevertices.Similarly,a tetrahedron
in R3 consistingof six rods connectedbut freely pivotingat the four vertices
is rigid while the one-skeletonof a cube in R3 is flexible. A figure consisting
of two triangleswith a commonedge is rigidin 12 but flexiblein R3since one
trianglecan then rotaterelativeto the otheralong the commonedge.
Severalkinds of physicalproblems,includingthe one just described,share
the followingmathematicaldescription.Considera finite set V of points in RW
togetherwith a collectionE of pairs of points in V, which is to be thoughtof
as the set of pairs of points that are connected.A continuoustime dependent
transformationof the points in V is a flexing of the structureif the distances
betweenpairs of points in E remainfixed in time but the final configuration
is not congruent(in the Euclideansense) to the originalconfiguration.If no
flexingexists, the structureis said to be rigid.
When the problemis formulatedin this way, the usefulnessof the language
of graph theory becomes apparent. For example, a disconnected graph
embeddedin R' is flexiblein RWwhereasa completegraphembeddedin RWis
rigid in R'. However, the rigidityor flexibility of a graph embeddedin RW
cannot be determinedsimplyfrom the abstractstructureof the graph,for it is
Receivedby the editorsFebruary24, 1977and, in revisedform,May 16, 1977.
AMS (MOS) subject classifications (1970). Primary 05C10, 50B99, 52A25, 70B15.
? American Mathematical Society 1979
279
280
L. ASIMOW AND B. ROTH
not difficult to find a graph which when embedded in RWin one way is
flexibleand in anotherway rigid.(See Examples1, 2, 3 of ?5.)
In this paperwe presentcriteriafor determiningwhethera particulargraph
will be rigid or flexible in RWfor almost all locations of its vertices, where
"almost all" has both a topological and a measure theoretic meaning.
Furthermore,we find that a graphin R' is eitherrigidfor almostall locations
of its verticesor flexiblefor almostall locationsof its vertices.
Throughoutthis paper, our interestfocuses on a notion of rigiditythat is
sometimes referred to as "continuous rigidity". The related concept of
"infinitesimalrigidity"which is not discussedin this paperwill be dealt with
in a sequel.In the literature,the term"rigid"is used in both of these sensesas
well as severalothers.In this paper,"rigid"is alwaysmeantin the continuous
sense that is discussedinformallyhere in the introductionand definedin ?2.
Much of the presentpaper was inspiredby HermanGluck'spaper [2] on
rigidity. We are also grateful to Branko Grunbaumfor providing us with
severalinterestingexamples.
2. Preliminaries.For our purposes,an (abstract)graphG is to be thoughtof
as a set V = {1, 2, . . . , v} togetherwith a nonempty set E of two-element
subsets of V. Each element of V is referredto as a vertex of G and each
elementof E is called an edge of G. On the otherhand, a graphG(p) in Rn is
a graph G = (V, E) together with a point p = (p1,
p,) E RV ) . . . x
R' = RIV. We refer to the points pi for i E V as the vertices and the line
segments[pi,pj] in R' for {i, j) E E as the edges of the graph G(p) in RW.
Note that for a graphG(p) in R', we have not requiredthatpi 7# pj for i :# j
and thus it is surelyinappropriateto speak of G(p) as an embeddingof G in
R .
Consider a graph G = (V, E) with v vertices and e edges, that is, V-
{ 1, ... , v} and E has e elements. Order the e edges of G in some way and
definefG: Rnl -* R' by
fG(tl***At)
_
= (***Ilti
tj|2..
where {i,j} E E, tkE Rn for 1 < k < v, and 11 1 denotes the Euclidean
norm in RW.Note that if G(p) is a graph in Rn, thenfG(p) E R' consists of
the squaresof the lengthsof the e edges of G(p) and thus we referto fG as the
edge function of the graph G. If
fG(p)
=
fG(q) for p, q E Rnv, then the
correspondingedges of the graphs G(p) and G(q) in RWhave the same
length. Consequently, the structure near p of the real algebraic variety
fJ '(fG(p)) is pertinentto the determinationof the rigidityor flexibilityof the
graphG(p) in RW.
Let KV(or simply K) denote the complete graph with v vertices, which
means that every two-element subset of V = { 1, ... , v} is an edge of Kv.
THERIGIDITYOF GRAPHS
281
Note that fK(p) = fK(q) for p, q E RV if and only if the map pi -qi
1 S i S v, extends to an isometryof R'. Recall that an isometryT of R' is a
map T: R' -* R' such that Tx - Tyll = lx- y for all x, y E RW.Since the
set 4 (n) of isometries of RWis a smooth manifold, fA7-'(fK(p)) is also a smooth
manifold. (See ?3 for details.) If G is a graph with v vertices and K the
completegraphwith v vertices,then clearlyfy-'(fK(p)) c fJ '(fG(p)) and it is
the naturenearp of this inclusionthat determinesthe flexibilityor rigidityof
the graphG(p) in RW.
Next, we providedefinitionsof rigidityand flexibility.
DEFINITION.Let G be a graphwith v vertices,K the completegraphwith v
vertices, and p E Rn. The graph G(p) is rigid in RWif there exists a
neighborhood U of p in RWVsuch that
fK (fK (P)) n U = f
n U.
The graphG(p) is flexible in RWif there exists a continuouspath x: [0, 1]
RfnVsuch that x(0) = p and x(t) E fJ '(fG(P)) - fA7'(fK(p))
for all t E (0, 1].
Thus G(p) is rigid in RWif and only if for every q sufficientlyclose to p
withfG(q) = fG(p), there exists an isometryof RWtakingpi to qi for 1 S i S
v. On the other hand, G(p) is flexible in RWif and only if it is possible to
continuously move the vertices of G(p) to noncongruentpositions while
preservingthe edge lengthsof the figure.
In additionto establishingthe equivalenceof nonrigidityand flexibility,the
following proposition demonstratesthe equivalence of another reasonable
notion of flexibility:
I(fG (p))
PROPOSITION1. Let G be a graph with v vertices, K the completegraph with v
vertices, andp E R"nv.Thefollowing are equivalent:
(a) G(p) is not rigid in RW;
(b) G (p) is flexible in RW;
(c) there exists a continuouspath y in fJ '(fG(p)) with y (0) = p and y (t) M
f l (fK(p)) for some t E (0, 1].
If G(p) is not rigid in RW,then every neighborhoodof p contains
PROOF.
points of the algebraicvarietyfj- (fG(P)) not belonging to the subvariety
and thus the existence of an analytic path x with x(O) = p and
fk-'(fK(P))
0 < t S 1, follows from [5, Lemma 18.3].
x(t) E fj (fG(P)) - fki (f(P))A
(See also [4].)Thus (a) implies(b). Clearly(b) implies(c).
If (c) holds, then there exists to E [0, 1) such thaty (to) is the last point in
qv) andp = (pl, * * ,pv).
Then there is an isometry T of RWwith Tqi=pi, 1 S i S v, and thus
Since
(T o Yi' ... , T o yv) maps (to, 1] into f' (fG(P)) - fAY'(fK(P))o
o
=
.
.
T
of
intersects
p
fJ
(T y1, *,
'(fG(p))
yv)(to) p, every neighborhood
f,-7(fK(P))
-fk-
as t increases. Lety(t0) = q = (ql,
'(fK(p))
*.
,
and thereforeG(p) is not rigidin RW.
282
L. ASIMOW AND B. ROTH
For a smooth map f: X -* Y where X and Y are smooth manifolds,we
denote the derivative of f at x E X by df (x). Let k = max{rank df (x):
x E X }. We say that x E X is a regular point of f if rank df (x) = k and a
singularpoint otherwise.
PROPOSITION 2. Let f: Rn -> Rm be a smooth map and k
max{rank df (x): x E Rn}. If xo E Rn is a regular point of f, then the image
underf of some neighborhoodof xo is a k-dimensional manifold.
-
PROOF.
Let f = (f1,f2) wherefi consists of the first k coordinatefunctions
of f and assume that rank df1(xo) = k. Since rank df1 = k in a neighborhood
of xo, the InverseFunction Theoremyields local coordinatesat xo such that
Thus in local coordinates
fl(XII x2) = xl.
df4=
I
0
f2
8f2
ax,
.
aX2 J
Since rankdf = k near xo, af2/8x2 = 0 near xo. Hencef2(x1,x2) = g(x ) and
thereforef(x1, x2) = (x1, g(x)) near xo. Thusf maps some neighborhoodof
xo onto the graphof g whichis a k-dimensionalmanifold.
It follows that if p is a regularpoint of fG, thenfj-'(fG(p)) is a manifoldof
co-dimensionk nearp. In this case, the constructionof a smooth path in (a)
implies (b) of Proposition1 is straightforwardsince nearp we then have that
fA7'(fK(p)) is a propersubmanifoldof f- '(fGG(p)).
Finally, a subsetM of RWis said to be an affine set if M containsthe entire
line througheach pair of distinctpoints in M. The dimensionof an affine set
M in RWis defined to be the dimension of the subspace M
-M=
{x
-
y: x, y E M } parallelto M and the affine hull of a set S c Rnis the smallest
affine set containing S. For p
dimension of the affine hull of {p,
p
(p1, .. . pv) E Rnv, let dim p be the
* * p4
=
3. The main result.
THEOREM.Let G be a graph with v vertices, e edges, and edge function fG:
RnV-- Re. Suppose that p E RnVis a regular point of fG and let m = dim p.
Then the graph G (p) is rigid in Rnif and only if
rank dfG(p) = nv
-
(m + 1)(2n -m)2
and G (p) is flexible in Rnif and only if
rank dfG(p) < nv
-
(m + 1)(2n
-
m)/2.
PROOF. Let k = max{rank dfG(x): x E Rnv}. Then k = rank dfG(p). By
Proposition2 of ?2, there exists a neighborhood V of p in RfV such that
fJ-I(fG (p)) n V is an (nv - k)-dimensionalmanifold.
THE RIGIDITY OF GRAPHS
283
Let I (n) be the n(n + 1)/2-dimensionalmanifoldof isometriesof RWand
RflVby F(T) = (Tp1,...,
TpI) for T E 5 (n). Note that F
is smooth and im(F) = fiK'(fK(p)) where K is the complete graph with v
vertices. Let M be the affine hull of {p, ... ,p.} Thus dim M = dimp =
define F:
(n)
-*
m. Clearly F - l(p) is the subgroupof 4 (n) consisting of isometrieswhich
equal the identity on M and F- I(p) can be identified with the (n
-
m)(n
-
m - l)/2-dimensional manifold 6 (N) of orthogonallinear transformations
of N where N is the (n - m)-dimensionalsubspace orthogonal to the mdimensionalsubspaceM - M. Let
-
IT: 4(n)
(n)IF'(p)
be the naturalprojectionand define
F: 4(n)/F-l(p)
-Rnv
so that F = F o v. Then F is smooth since F is smooth and, moreover, F:
f (n)/F - '(p) -* im(F) is a diffeomorphism. Since 4 (n)/F - '(p) is a manifold
of dimension
n(n + 1)/2
-
(n
-
m)(n
-
m
-
= (m + 1)(2n -m)2,
1)/2
we conclude that im(F) = im(F) = fS '(fK(p)) is an (m + 1)(2n -m)2dimensionalmanifold.
The (m + 1)(2n - m)/2-dimensional manifold fi-(fK(p)) n v is a
submanifoldof the (nv - k)-dimensionalmanifoldfj '(fgG(p))n v. Therefore we have
k < nv
-
(m + 1)(2n
-
m)/2.
Clearly k = nv - (m + 1)(2n - m)/2 if and only if there exists a neighborhood U of p in RfV such thatfK I(fK(p)) n U = fj I(fG(p)) n U, thatis,
if and only if G(p) is rigid in RW.Since k < nv - (m + 1)(2n - m)/2, G(p)
is flexible in RWif and only if k < nv - (m + 1)(2n - m)/2, which con-
cludes the proof.
Let P (x) be the sum of the squaresof the determinantsof all the k x k
submatricesof dfG(x) for x E RnV. Then P is a nontrivialpolynomialin nv
variablesand thus the set R of regularpoints of fG is a dense open subset of
RfV since R = {x E Rnv : P(x) 7# O}. Moreover,Fubini's Theorem enables
one to conclude that the set
RfV -
R of singular points of fG has Lebesgue
measurezero in Rnv. Thereforethe above theoremdeterminesthe rigidityor
flexibilityin RWof a graph G(p) for almost all p E Rnv, where "almostall"
can be interpretedboth topologicallyand measuretheoretically.Furthermore,
in ?4 we show that this determinationis constantfor regularpoints of fG, that
is, eitherG(p) is rigidin RWfor allp E R or G (p) is flexiblefor allp E R.
4. Corollaries.In the following lemma and our first two corollaries,the
dimensionn of the Euclideanspace in which we considera graphvaries, so
L. ASIMOW AND B. ROTH
284
we temporarily denote the edge function of a graph G by fG,n. Thus fG,n:
RnV- R' and
fG,n
(tl
*e
tv)
=(**eeli-
j||2
**
where {i, j} is an edge of G and each tk E Rn.
LEMMA.Let G be a graph with v vertices. Supposep E RfV is a regularpoint
of fG,n and let m = dim p. Then there exists q E Rrv such that q is a regular
point of fG,m' dim q = m, and rank dfG,(q) = rank dfG,n(p). Moreover, if
G (p) is rigid in Rn, then G (q) is rigid in Rm.
Define C: Rm -* RW by C(xI, . .. , x,) = (xl, .. ., xm 0, ... . 0).
There exists an isometry T of Rn taking the m-dimensionalsubspaceim(C)
onto the affine hull M of {p1,I ... ,pv. Then T o C maps Rm onto M and it
is not difficult to show that for all (tl, . .. , tv) E Rmv,
PROOF.
rank dfG,m(t1,....,
tv) = rank dfGfn (TCtl
., TCtv).
...
Choose q = (q1, ... , qv) E Rrv such that TCqi = pi, 1 < i < v. Clearly
dim q = dim p and max{rank dfG,m} < max{rank dfG,n} = rank dfG,n(P) =
rank dfG,m(q) < max{rank dfG,m}
. Thereforeq is a regularpoint of fG,m.
If G(q) is flexible in Rm, then the flexing of G(q) guaranteedby the
definitiontogetherwith T o C providesa flexingof G(p) in RW.
Considera graph G with v vertices.Then RlV is partitionedin three ways.
Thereis the set whereG(p) is rigidin RI and the set whereG(p) is flexiblein
Rn; there are the sets of regularand singularpoints of the edge functionfG;
and there are the sets where dimp = min{v
-
1, n} and dimp < min{v
-
1, n}. How are these three partitionsrelated?Our first corollaryestablishes
that it is neverthe case that G(p) is rigidin Rn wherep is a regularpoint of fG
with dimp < min{v - 1, n). It is not difficult to find graphs G and points
p E R2Vshowingthe seven remainingpossibilitiescan all occur.
COROLLARY
1. Let G be a graph with v vertices. If G (p) is rigid in Rn where
n}.
p E Rnvis a regularpoint offkf, then dimp = min {v -,
PROOF.Let m = dim p. By the lemma, there exists q E R"' with q a regular
point of fG,m, dim q = m, rank dfG,(q) = rank dfG,n(p), and G(q) rigid in
Rm.Applyingthe theoremof ?3 to G(p) in Rnand G(q) in Rm,we obtain
mv
-
(m + 1)(2m
-
m)/2
= rank dfG,m (q) =
= nv
-
rank dfG,n(P)
(m + 1)(2n
-
m)/2.
Since the function g(x) = vx - (m + 1)(2x - m)/2 is affine and g(m) =
g(n), either m = n or the coefficient v - (m + 1) of x in g equals zero.
Therefore m = min{v - 1, n}.
In connection with Corollary 1, we note that if dimp = v
-
1, then for
THE RIGIDITY OF GRAPHS
285
eachj the set {pi - p: i # j} is linearlyindependent.It follows thatp is a
regularpoint since rankdfG(p) = e.
Let G be the graphconsistingof two trianglessharinga commonedge and
letfG: R2
->
Rs be the edge function of G in R3. Then it is a simple matter to
find regularpointsp,q E R12 of fG where dimp = 3 and dim q = 2. In this
case, G(p) and G(q) are both flexiblein R3iand our next corollarystates that
regularpoints of different dimensionscannot occur if one of the graphs is
rigid.
2. Let G be a graph with v vertices and edge function fG,n:
Re. If p, q E RnVare regularpoints OffG,n and G (p) is rigid in RI, then
G (q) is also rigid in Rn and dim p = dim q.
COROLLARY
RfnV--
PROOF.
Let m = dimp and 1 = dim q. By Corollary 1, m = min{v
1, n}
-
and, as always, 1 S min{v - 1, n}. Sincep and q are both regularpoints of
fG,n the lemma givesp' e Rmvand q' E RlVwithp' a regularpoint of fG,m q' a
regular point of fGl, dim p' = m, dim q' = 1, and rank dfGm(P') =
rank dfG,n (P) = rank dfG,n(q)= rank dfG,1(q'). By the theorem of ?3,
rank dfG1(q') S lv - 1(1 + 1)/2. By the lemma, G(p') is rigid in Rm and thus
rank dfG,m(P') = mv - m(m + 1)/2 by the theorem. But the function g(x) =
vx - x(x + 1)/2 is strictly increasing for x S v - 1 and thus 1 = m since
g(l) = g(m). Therefore dimp = dim q which implies that G(q) is also rigid
in R .
Since Corollary2 guaranteesthat G(p) is rigidin RWfor all regularpointsp
wheneverit is rigid for a single regularpoint, we can conclude that either
G(p) is rigid in RWfor all regularpoints or G(p) is flexible in RWfor all
regularpoints.
Our next corollarymakes precise the appealingidea that a graphwith too
few edges can almost never be rigid. (See ?5 for examples of rigid graphs
havingtoo few edges.)
COROLLARY 3. Let G be a graph with v vertices and e edges. If e < nv
+ 1)/2, then G (p) is flexible in RWfor all regularpoints p of fG.
PROOF.
-
n(n
Letp E R'v be a regularpoint of fG and m = dimp. Then
rank dfG(p) < e < nv
-
n(n + 1)/2 < nv
-
(m + 1)(2n
-
m)/2
and thus G(p) is flexiblein RWby the theoremof ?3.
Corollary3 representsa first step in the directionof a purelycombinatorial
method for determiningwhethera graphis almost always rigid or flexible in
Rn. A graphtheoreticcharacterization
in the case n = 2 is given by Laman[3,
Theorem6.5].
Let K be a complete graph. By the definition of rigidity,it is clear that
K(p) is rigid in RWfor all n and all p E Rnv. One consequenceof our next
286
L. ASIMOW AND B. ROTH
corollaryis that an incompletegraph is almost always flexible in RWfor all
n > v - 1.
4. Let G be a graph with v vertices. Thefollowing are equivalent:
COROLLARY
(a) G is a complete graph;
(b) for all n, G(p) is rigid in R for allp E R"';
for all regularpoints p.
(c) for some n > v - 1, G (p) is rigid in RW
PROOF.
The fact that (a) implies (b) is a consequenceof the definition of
rigidity and (b) obviously implies (c). Suppose there exists n > v - 1 such
that G(p) is rigid in RWwherep E RnVis a regularpoint. By Corollary 1,
dimp = v - 1 and thus rank dfG(p) = v(v - 1)/2 by the theorem of ?3. But
e < v(v - 1)/2 = rank dfG(p) < e and therefore e = v(v - 1)/2, that is, G
is a completegraph.
Our last two corollariesconcernplanargraphs,which are graphsthat can
be embeddedin the plane R2, that is, drawnin the plane in such a way that
edges intersectonly at the appropriatevertices.For a connectedplanargraph
G, Euler'sformula implies that the numberf of faces of G is well-defined
since v - e + f = 2 for any embedding of G in the plane. For a connected
planargraph,we define the averagenumberA of edges per face by A = 2e/f.
A version of our next corollarywas first suggestedand proved by Sherman
Stein.
COROLLARY5. Let G be a planar graph such that G (p) is rigid in R2for all
regularpoints p of fG. Then the average numberA of edges per face of G is less
thanfour and if G has more than two vertices, then G contains a triangle.
PROOF.Since G (p) is rigid in R2 for all regular points p
connected and Corollary 3 implies that e > 2v - 3. Therefore
A = 2e/f = 2e/(2 - v + e) < 4e/(e + 1) <4.
E R2V,
G is
Now supposev > 3 and considerany embeddingof the connectedplanar
graphG in R2. Everyface of the embeddedgraphis boundedby at least three
edges, wherean edge is counted twice for a face if the face lies on both sides
of the edge. If no face has exactly threeboundaryedges, then 2e > 4f which
contradictsA = 2e/f < 4. Thereforesome face of the embeddedgraphhas
exactly three boundary edges which implies that G contains a subgraph
isomorphicto the graphK3,that is, G containsa triangle.
It can be shown that four is the best possiblebound in Corollary5.
If G is a connectedplanargraphwith v > 3, then 3f < 2e and 3f = 2e if
and only if every face (includingthe unboundedone) of some (or equivalently every) embeddingof G in the plane has exactly threeboundaryedges,
whereagain an edge is countedtwice for a face if the edge lies inside the face.
Thus we referto connectedplanargraphswith 3f = 2e as triangular.
THE RIGIDITY OF GRAPHS
287
We say that a graph is polyhedralif the graph can be embeddedin R3 in
such a way that its verticesand edges are the verticesand edges of a convex
polyhedron in R3. Thus G is polyhedral if and only if there exists p =
(Pi, . . , pE R3E such that pi4, pj for i #1j and the edges [pi,pj] of G(p)
are the edges of a convex polyhedronin R3.Clearlyevery polyhedralgraphis
3-connectedand planar.
If G is polyhedraland p E R3v is chosen so that the verticesand edges of
G(p) are the vertices and edges of a convex polyhedronin R3, then p is a
regular point of fG since rank dfG(p) = e. This fact stems from some
approachesto Cauchy'sTheorem on the rigidity of polyhedrain R3. (For
example,see [2, Lemma5.2, Lemma5.3, and the proof of Theorem5.1] where
we interpret"strictlyconvex"to mean edges as well as verticesare extremal.)
The following corollaryis essentiallya reformulationof resultsof Gluck [2]
obtainedin a somewhatdifferentsetting.
COROLLARY 6. Let G be a connectedplanar graph with v > 4. The following
are equivalent:
(a) G (p) is rigid in R3for all regularpoints p E R3v;
(b) G is triangular;
(c) G is triangularandpolyhedral.
PROOF.If (a) holds, then e > 3v
e > 3v
-
6 = 3(v
-
-
6 by Corollary 3. Thus
2) = 3(e -f)
= e + (2e - 3f) > e
and therefore2e = 3f whichmeansthat G is triangular.
The fact that (b) implies (c) follows from Steinitz'sTheorem [1], which
providesnecessaryand sufficientconditionsfor a collectionof vertices,edges,
and faces to be realized as the vertices, edges, and faces of a convex
polyhedron.It is here that we need v > 4.
Suppose (c) holds. Then there exists q E R3Vsuch that the vertices and
edges of G(q) are the verticesand edges of a convex polyhedronin R3 and
2e = 3f. Thus rank dfG(q)= e = 3(e-J) = 3(v -2) = 3v-6
which
impliesthat G(q) is rigid in R3by the theoremof ?3. ThereforeG(p) is rigid
in R3for all regularpointsp.
In particular,the one-skeletonof a convex polyhedronin R3is rigidin R3if
and only if every face of the polyhedronis a triangle.Moreover,the use of
the result on rank arising from Cauchy's Theorem in conjunction with
Corollary3 shows that for a convexpolyhedronin R3with all triangularfaces
the removalof any edge fromits one-skeletonleads to flexibility.
Another condition equivalentto (a), (b), and (c) in Corollary6 is "G is
triangularand 3-connected".Thereforeif G is a connectedplanargraphwith
v > 4 such that G(p) is almost always rigid in R3, then G is 3-connected.
More generally,the refereehas observedthat if G is a graphwith v > n + 1
288
L. ASIMOW AND B. ROTH
such that G(p) is rigidin RWfor all regularpointsp, then G is n-connected.
5. Examples.It is quite easy to induce pathologicalbehaviorby allowing
some of the vertices of a graph G(p) in RWto coincide; we have chosen
examplesin which this does not occur.
EXAMPLE
1. Let G be the graph shown in Figure 1 for which v = 6 and
e = 8. Since e < 2v - 3, Corollary3 impliesthat G(p) is flexiblein R2 for all
regularpointsp E R12. However,if the location q of the verticesis as shown
in Figure 1, then a simple geometricalargumentshows that G(q) is rigid in
R2. By perturbingthe four collinear vertices of G(q), one obtains graphs
isomorphicto G(q) but flexible in R2.We note also that the averagenumber
of edges per face of the rigidgraphG(q) equalsfour.
FIGURE
1
Our next example is almost always rigid in R2 and only occasionally
flexible,in contrastto Example1.
EXAMPLE
2. Let G be the graph shown in Figure 2 for which v = 6 and
e = 9. For the location q of the vertices shown in Figure 2, a simple
geometricalargumentshows that G(q) is flexible in R2. However, it is not
difficultto findppE R12 with rankdfG(p) = 9 and dimp = 2. ThereforeG(p)
is rigid in R2 by the theorem of ?3 and thus rigid at all regularpoints by
Corollary2.
FIGURE
2
EXAMPLE
3. Let G be the graph shown in Figure 3 which consists of a
tetrahedronwith a triangleinside its base. Since G is planarbut not triangular, G(p) is flexible in R3 for all regular points p E R2' by Corollary 6.
However, if the location q of the vertices is as shown in Figure 3 with the
triangle lying in the plane of the base of the tetrahedron,then a simple
geometricalargumentshows that G(q) is rigid in R3 even though G(q) has
threenontriangularfaces.
One consequence of Cauchy's Theorem is that if G(p) forms the oneskeleton of a convex polyhedron in R3, then rank dfG(p) = e. Our last
THE RIGIDITY OF GRAPHS
289
FIGURE 3
example shows that this fact does not generalize to higher dimensional
polytopes.
EXAMPLE
4. Let C be a convex polytope in RWwith nonemptyinterior.We
construct the convex polytope r(C) in Rn+' as follows. Embed C in a
+ and choose new vertices x, y E R +1 - H such that
hyperplane H in RW
the line segment [x, y] intersectsthe relativeinteriorof C. Then r(C) is the
convex hull of C with [x, y] and it can be verifiedthat the verticesof r(C) are
x and y togetherwith the verticesof C. Also the edges of r(C) are the edges
of C togetherwith all [x, z] and [y, z] wherez is a vertexof C. Thus, if v' and
e' are the number of vertices and edges of r(C), we have v' = v + 2 and
e' = e + 2v wherev and e are the numberof verticesand edges of C.
Now let C be a convex polygon in R2with v verticesand v edges. Then for
r(C) in R3, we have v' = v + 2 and e' = 3v, while for r2(C) in R4, we have
v" = v + 4 and e" = 5v + 4. Suppose G is the graph of vertices and edges of
r2(C). Then forv > 3,.
max{rank dfG} < 4v"
-
10 = 4v + 6 < 5v + 4 = e".
Finally,we note that for v = 3, if C, r(C) and r2(C) are suitablychosen and
G(p) forms the one-skeletonof r2(C), then rankdfG(p) = 18.Thereforep is a
regularpoint of fG and G(p) is rigid in R4.
REFERENCES
1. D. Barnetteand B. Griinbaum,On Steinitz'stheoremconcerningconvex3-polytopesand on
somepropertiesofplanargraphs,The ManyFacetsof GraphTheory,LectureNotes in Math.,vol.
Berlin,1969,pp. 27-40.
110,Springer-Verlag,
2. H. Gluck,Almostall simplyconnectedclosedsurfacesare rigid,GeometricTopology,Lecture
Berlin,1975,pp. 225-239.
Notes in Math.,no. 438, Springer-Verlag,
3. G. Laman, On graphsand rigidityof plane skeletalstructures,J. Engrg.Math. 4 (1970),
331-340.
Ann. of Math.Studies,no. 61, Princeton
4. J. Milnor,Singularpointsof complexhypersurfaces,
Univ. Press,Princeton,N.J., 1968.
of curves,Canad.J. Math. 10 (1958),242-278.
5. A. Wallace,Algebraicapproximation
WYOMNG 82701 (Current
DEPARTMENTOF MATHEMATICS,UNIVERSITYOF WYOMNG, LARAimE,
address of B. Roth)
Current address (L. Asimow): Department of Mathematics, Syracuse University, Syracuse,
New York 13210