## Journal of the Brazilian Computer Society

##
*Print version* ISSN 0104-6500*On-line version* ISSN 1678-4804

### J. Braz. Comp. Soc. vol.7 no.3 Campinas 2001

#### http://dx.doi.org/10.1590/S0104-65002001000200010

**ARTICLES**

**On the homotopy type of the clique graph ^{*}**

**F. Larrión ^{I,}^{1}; V. Neumann-Lara^{I,}^{1}; M. A. Pizaña^{II}**

^{I}Instituto de Matemáticas, U.N.A.M., Circuito Exterior, C.U. México 04510 D.F. MÉXICO., {paco, neumann}@matem.unam.mx

^{II}Universidad Autónoma Metropolitana, Depto. de Ingeniería Eléctrica., Av. Michoacán y Purísima s/n México 09340 D.F. MÉXICO., map@xanum.uam.mx, http://xamanek.uam.mx/map

**ABSTRACT**

If *G* is a graph, its clique graph *K(G) *is the intersection graph of all its (maximal) cliques. The complex *G* of a graph *G *is the simplicial complex whose simplexes are the vertex sets of the complete subgraphs of *G. *Here we study a sufficient condition for G and

*K(G)*to be homotopic. Applying this result to Whitney triangulations of surfaces, we construct an infinite family of examples which solve in the affirmative Prisner's open problem 1 in

*Graph Dynamics*(Longman, Harlow, 1995): Are there finite connected graphs

*G*that are periodic under

*K*and where the second modulo 2 Betti number is greater than 0?

**Keywords: **clique graphs, clique convergence, Whitney triangulations, clean triangulations, simplicial complexes, modulo 2 Betti numbers.

**1 Introduction and terminology**

All our graphs are simple. If G is a graph, a *complete *of *G* is a complete subgraph of *G* and a *clique *is a maximal complete of *G*. The clique number w(*G*) is the maximum order of a clique of *G*. We shall often identify induced subgraphs with their vertex sets. In particular, we shall often write *x *Î *G* instead of *x *Î* V*(*G*).

We say that G is *locally H *if the subgraph *N _{G}*(

*x*) induced in

*G*by the (open) neighbourhood of any vertex a; Î

*G*is isomorphic to

*H.*We say that

*G*is

*locally , =*{

*H*

_{1}

*,H*

_{2}

*,...*} if for every

*x*Î G,

*N*(

_{G}*x*) @

*H*

_{i}for some

*H*

_{i}6 . C

_{n}and

*P*are, respectively, the cyclic and path graphs on

_{n }*n*vertices. We say that G is

*locally cyclic*if it is locally {

*C*}.

_{n}:n__>__3 The *clique graph K*(*G*) of G has all cliques of G as vertices, two of them being adjacent iff they (are different and) share some vertex of G. We call *K *the *clique operator. *Iterated clique graphs are inductively defined by *K*^{0}(*G*)* = G* and *K ^{n+1}*(

*G*)

*= K*(

*K*(

^{n}*G*))

*. G*is

*K-periodic*if G @

*K*(

^{n}*G*) for some

*n*1. Extensive bibliography on clique graphs can be found in [14].

__>__ A graph *G* is clique-Helly if whenever *X = *{*q _{i},...,q_{n}* } Í

*V*(

*K*(

*G*)) is a family of pairwise intersecting cliques, then

*X*¹ Æ. We say that

*Q =*{

*q*

_{1}

*,...,q*} Î

_{n}*V*(

*K*(

^{2}*G*)) is a

*star*of

*G*if Q ¹ Æ, otherwise it is a

*necktie*of

*G*. Obviously, a graph is clique-Helly iff it has no necktie.

If *G* is a graph, *G* is the simplicial complex whose simplexes are the completes of G. We say that two simplicial complexes are homotopic ( ) when their geometric realizations are homotopic (|| ||). The behaviour of topological invariants of G under several graph operators (including the clique operator) has been studied in [9, 10, 11]. In particular, Prisner proved in [10] that if *G* is clique-Helly, *G* (*G*). Our main result (Theorem 2.4) states that this is also true for many non-clique-Helly graphs. As an application of this, we will show (Theorem 2.5) that if G is free of tetrahedra and induced octahedra, then *G* *K*(*G*) .

An interesting particular case is when the realization || is a compact surface (with or without border), i.e. is a triangulation of a compact surface. If *G* is the underlying graph (or 1-skeleton) of a surface triangulation , every face of is a triangle of *G *but the converse may not be true. We shall be interested in surface triangulations where every triangle of G is a face of : such a triangulation is a *Whitney triangulation *[17]. Thus, if is Whitney it is determined by *G, *and we tend to identify *G *with , and sometimes even with ||. If is Whitney, (except for the tetrahedron *K*_{4}) the cliques of G are precisely the faces of the triangulation. Whitney triangulations have other names and have been studied before [2, 5, 8, 16, 17]. In particular, the description of the dynamical behaviour under the clique operator of the regular Whitney triangulations has been completed in [8]. As a corollary to our Theorem 2.5, we will have that the only Whitney triangulation of a compact surface which is not homotopic to its clique graph is the octahedron. We shall use the following two theorems:

**Theorem 1.1 **[8] *G is the underlying graph of a Whitney triangulation of a closed surface (resp. compact surface) if and only if G is locally cyclic *(*resp. G is locally *{*C _{n},P_{m} *:

*n*3,

__>__*m*2}).

__>__ **Theorem 1.2 **[8] *For every Whitney triangulation G of a closed surface with minimum degree at least 7 we have K*(*G*) * K ^{3}*(

*G*).

We refer to [1], [10] and [13] for undefined concepts.

**2 Homotopy**

If . is a hypergraph, * denotes its dual hyper-graph, and ^{¯} is the smallest simplicial complex containing the hyperedges of as simplexes. The following reformulation is due to Prisner [9, 10]:

**Theorem 2.1 **(Dowker, [3]) *For every hyper-graph* , ^{¯}* and* ^{*¯}* are homotopic.*

If *G* is a graph, (*G*) is its *clique hypergraph:* (*G*) has the same vertex set as *G* and its hyper-edges are the cliques of *G*. It follows immediately from the definitions that *G* = (*G*)^{¯}.

The *star hypergraph S*(*G*) of *G* has the same vertex set as *K*(*G*) and the hyperedges are the cliques *Q = *{*q*_{1}*,q*_{2}*,...,q _{r}* } of

*K*(

*G*) satisfying Q ¹ Æ. It follows that (

*G*)

^{*¯}=

*S*(

*G*)

^{¯}and that

*S*(

*G*)

^{¯}Í

*K*(

*G*). The equality

*S*(

*G*)

^{¯}=

*K*(

*G*) holds precisely when

*G*is clique-Helly.

Then, as pointed out by Prisner [10, Proposition 2.2], it follows from Dowker's theorem that *G* and *K*(*G*) are homotopic for every clique-Helly graph *G*. A reformulation of this result will be useful to us:

**Theorem 2.2 **(Prisner, [10]) *For every graph G, we have G* = (*G*)^{¯} (*G*)^{*¯} = *S*(*G*)^{¯} Í *K(G*)*. In particular, if G is clique-Helly, then G* *K*(*G*).

Prisner provided examples of graphs *G* (namely the n-dimensional octahedra, for *n > *3) such that

*G*and

*K*(

*G*) are not homotopic. As we shall see shortly, this property of the octahedra is tightly connected to the fact that octahedra contain neckties without a center.

**Definition 2.3 ***If X is a complete of K*(*G*)* satisfying X ¹ Æ, then q _{0 }Î K*(

*G*)

*is called a center of X if:*

*Y *Í* X *and *Y *¹ Æ imply (*Y *È{*q*_{0}}) ¹ Æ .

Note that *X *È {*q*_{0} } is always a complete of *K*(*G*). Also, when such an *X *is a clique of *K*(*G*)*, X *must contain all its centers.

Many non-Helly graphs *G *satisfy *G* *K*(*G*), Indeed we shall show that for many non-Helly graphs *G, S*(*G*)^{¯} is a strong deformation retract *of K*(*G*).

Let's rename = *S*(*G*)^{¯} and = *K*(*G*). We know that Í . Note that the 0-simplexes of and are the same. In order to easily define the required mappings, we take the barycentric subdivision ' of relative to as used in [12, page 19].

Equivalently, we define the complex ' whose vertices are those of (denoted by *q _{i}*) plus a (formal) barycenter

*b*(

*s*) for each

*s*Î – , and whose simplexes are of the form {

*q*

_{1}

*,...,q*(

_{n},*s*

_{1})

*,..., b*(

*s*)} and satisfy:

_{m}1. {*q*_{1}*,...,q _{n}*}Î.

2. *s _{j} *Î – for all

*j.*

3. *s _{i} *Î

*s*

_{1}for all

*i.*

4. *s _{j} *Í

*s*

_{j}_{+1}for ali

*j.*

and then we may prove that this is indeed a subdivision of using Theorem 3.3.4 in [13]. Of course, we still have Í '.

The idea behind this is to grab the offending simplexes (those in – ) by its barycenters and retract them into S. Now we can prove our main result:

**Theorem 2.4 ***Let G be a graph. Assume that any complete X of K*(*G*)* with * *X =* Æ* has a center which belongs to every necktie containing X. Then S*(*G*)^{¯} * is a strong deformation retract of K*(*G*). In particular, *G* K(*G*).

**Proof. **For every simplex *s* in – select, using the hypothesis, a fixed center *q*(*s*) of *s *belonging to every maximal simplex (i.e. necktie) that contains *s. *Also, for each *s *Î – , define

*s = *{*Q *Î *K*^{2} (*G*) : *s *Í *Q*}

Note: *s* Í Î – , and *s *Í* s' *implies *q*() Î Í .

Now define the map j_{1} : ' ® by j_{1} (*q _{i}*) =

*q*and j

_{i}_{1}(

*b*(

*s*))

_{j}*= q*(

*). Then for any simplex of ' we have that j*

_{j}_{1}{

*q*

_{1}

*,...,q*(

_{n},*s*

_{1})

*,..., b*(

*s*)}

_{m}*=*{

*q*

_{1}

*,...,q*, (

_{n}_{1})

*,..., b*()}. This is a simplex of because there is a clique

*Q*of

*K*(

*G*) such that

*q*Î

_{i}*Q*and

*s*Í

_{j}*Q*for all

*i = l,...,n*and

*j =*1

*,..., m*(take a

*Q*with

*s*Í

_{m}*Q*). Therefore

*q*(

_{i},q*) Î*

_{j}*Q*for all

*i*and

*j.*It follows that j

_{1}

*:*' ® is a Simplicial map, so | j

_{1}| : |'| ® || is continuous.

We claim now that Im(j_{1} i) = : As *q*_{1} Ç ¼ Ç *q _{n} *¹ Æ and {

*q*

_{1}

*,...,q*} Í

_{n}*s*

_{1}Í

_{1}, we obtain that

*q*

_{1}Ç ¼ Ç

*q*(

_{n}q_{1})¹ Æ. Using that

_{1}Í

_{2}Í ¼ Í and

*q*(

*) Î*

_{j}*for all*

_{j}*j,*it follows by induction that {

*q*

_{1}

*,...,q*, (

_{n}_{1})

*,..., b*()} is a simplex of . Now we know that Im(|j

_{1}| ) = || and that the restriction of |j

_{1}|to || is the identity in ||.

On the other hand, consider the canonical homeomorphism j_{0} : || ® |'|. Let j *= *j_{1}* o *j_{0}. Note that for all *x *Î || there is a simplex s Î such that *x, *j(*x*) Î |s| (any maximal simplex s Î satisfying x Î |s| will do). Then it follows that j 1 via the homotopy *H*(*x, t*)* tx + *(1 – *t*)j(*x*) (see, for example [6, Prop. 1.7.5]). Since j| = 1, we have that *H*(*x,t*)* = x *for all *x *Î ||. Therefore || is a strong deformation retract of ||.

An interesting consequence is the following:

**Theorem 2.5 ***If G is a graph free of induced octahedra and w*(*G*) * <* 3,

*then G*

*K*(

*G*)

**Proof. **Without loss of generality we assume *G *to be connected and non-trivial. Then we observe that every clique of G is a triangle or an edge.

Let *X *be a complete of *K*(*G*) satisfying *X = *Æ*, *and let *Z = *{*q _{1},...,q_{r} *} be a minimal subset of

*X*also satisfying Z = Æ.

Since *Z *is minimal and necessarily *r >* 3, we may take

*x*

_{23}Î (

*Z *{

*q*})

_{1}*, x*(

_{13}Î*Z *{

*q*

_{2}}) and

*x*

_{12}Î (Z – {

*q*

_{3}})

*-*Hence,

*q*{

_{0}=*x*

_{12},

*x*

_{13},

*x*

_{23}} is a clique of

*G.*This very construction was used by J. L. Szwarcfiter in his celebrated characterization of clique-Helly graphs [15].

It follows that *q*_{1} * = *{*x*_{12}, *x*_{13} , *a*}, *q*_{2}* = *{*x*_{12}, *x*_{13} , *b*} and *q _{3} = *{

*x*

_{13},

*x*

_{23},

*c*} for some three (different) vertices

*a*,

*b*,

*c*Î

*G*. Since

*q*

_{1}Ç

*q*

_{2 }Ç

*q*Æ it follows that

_{3}=*Z =*{

*q*

_{1}

*, q*

_{2}

*, q*}.

_{3} Let *Q *Î* K ^{2}*(

*G*) be a necktie containing

*Z,*and let

*q*Î

*Q.*If

*q*n

*qo =*Æ, then

*q =*{

*a, b, c*} and the set of vertices {

*x*

_{12},

*x*

_{13},

*x*

_{23}

*, a, b, c*} induces an octahedron in

*G*, contradicting our hypotheses. If |

*q*Ç

*q*|

_{0}*=*1

*,*say

*q*n

*q*{

_{0}=*x*

_{12}}

*,*then

*q*Ç

*q*

_{3}

*=*{

*c*} and {

*x*

_{12},

*x*

_{13},

*x*

_{23}, c} would contradict

*w*(

*G*)

*3. Therefore |*

__<__*q*Ç

*q*|

_{0}*2 for every*

__>__*q*Î

*Q.*

Since the set {*q * Î *K*(*G*) : |*q *Ç *q _{0}*|

*2} is a complete of*

__>__*K*(

*G*) it follows that

*Q =*{

*q*Î

*K*(

*G*) : |

*q*Ç

*q*|

_{0}*2}. Now the condition on the clique number implies that*

__>__*q*is a center of

_{0}*Q.*Then

*Q*is the unique necktie containing Z, so it is also unique containing

*X.*Therefore go is a center of

*X*which belongs to every necktie containing

*X,*and we apply the previous theorem.

The following result is an immediate consequence:

**Corollary 2.6 ***The only Whitney triangulation of a compact surface (with or without border) which is not homotopic to its clique graph is the octahedron. *

Now let's denote the *i-ih *modulo 2 Betti number of a complex by * _{i}*(). Take any locally {

*C*} graph

_{t}: t__>__7*H.*By Theorem 1.1 if is a Whitney triangulation of a closed surface, so we have

_{2}(

*H*) = 1. Since

*H*

*K*(

*H*) we have

_{2}(

*K*(

*H*))

*= I.*Then Theorem 1.2 tells us that

*G := K*(

*H*) is .K''-periodic, thus solving Prisner's open problem 1 in [11].

As a concrete example, it is shown in [8] that *I* ´ *K*_{3} is a locally *C*_{10} graph (here *I* is the icosahedron and {(*a, b*), (*a*', *b*')} Î *E *(*A* ´ *B*) iff {*a, a*'} Î *E *(*A*) and {*b, b'*} e *E*(*B*) ). In fact, Brown and Connelly [2] proved that for every *t *there is at least one finite locally *C _{t} *graph. Next, we shall construct an explicit infinite family of locally

*C*

_{7}graphs.

**3 Whitney triangulations**

Let's start with an infinite graph *T:* *V*(*T*) = Å and put *N = *{±(1,0),±(0,1),±(1,-1)}, then define {*x, y*} Î * E*(*T*) if and only *if y – x *Î *N.*

Each vector *u *Î Å gives rise to a translation *x u + x *which is an automorphism of *T. *Every finite locally *C _{6} *graph triangulating the torus is a quotient

*T/*G where G is the translation group generated by the translations given by two linearly independent vectors

*u, v*Î Å . The group G must satisfy the following admissibility condition: for every g Î G and

*v*Î

*V*(

*T*)

*,*the distance in

*T*from

*v*to g(

*v*) is at least 4 (otherwise, the resulting triangulation is not Whitney, see [7]).

Let *u = *(4,1), fix *r > *2, and let

*v*(2r,4r). Let G

_{T}=_{r}be the translation group defined by

*u*and

*v*and let

_{r},*be the parallelogram defined by these two vectors. The locally*

_{r}*C*graph

_{6}*G*G

_{r}= T/_{r}defines a Whitney triangulation of the torus with 14

*r*vertices:

*G*is obtained by identifying the parallel edges of

_{r}

_{r}. Now consider the 2r vertices *w*_{1}*, w*_{2}*, w*_{2}* _{r }*of

*G*which correspond to the vertices (2,1),(3,3),...,(2

_{r}*r*+ 1,4

*r*- 1) in

*i.e.*

_{r},*w*

_{i}

*=*(

*i +*1,2

*i *1). The vertices of

*G*are the disjoint union of the closed neighbourhoods

_{T}*N*[

*w*

_{i}] of these vertices, and removing these vertices from

*G*we obtain a locally

_{r}*P*

_{5}graph of order 12

*r*. Let us call

*the surface triangulated by , which is a torus with 2*

_{r}*r*open disks removed. All the vertices of lie in the border of

*The connected components of the border of*

_{r}.*are the hexagons*

_{r}*H*

_{1}

*, H*

_{2}

*,*...,

*H*

_{2}

*which were the open neighbourhoods of the removed vertices*

_{r}*w*

_{1}

*, w*

_{2}

*,..., w*

_{2}

*of*

_{r}*G*

_{r}**.**

Consider the locally *P*_{4} graph in figure 3. This graph gives us a Whitney triangulation of a cylinder, all the vertices lie in the border whose connected components *B*_{1 }and *B*_{2} are induced hexagons of .

Now, take the surface * _{r} *(with its Whitney triangulation given by the graph ) and

*r*different copies of the cylinder (with the Whitney triangulation given by). For the first copy, identify

*B*

_{1}with

*HI*and

*B*with

_{2}*H*

_{r+1}in an orientable manner, so a handle is glued to

*S*For the second copy, identify

_{r}.*B*

_{1}with

*H*and

_{1}*B*with

_{2}*H*so a second handle is glued to

_{r+1},*Continuing in this way, we obtain at the end a closed surface which is a sphere with*

_{r}.*r +*1 handles. The graph

*obtained from and the*

_{r}*r*copies of by the above method has 12r vertices and is the

*1-*skeleton of a triangulation of our surface .

As we want * _{r} *to be locally

*C*we have to take care so that the triangles in

_{7}*are exactly the triangles already present (16*

_{r}*r*inand 12 in each copy of ). This fails when two vertices

*x*Î

*H*and

_{i}*y*Î

*H*with

_{i+r}*d*(

*x,y*) < 3 in are identified with adjacent vertices in the

*i*-th copy of . Since

*d*(

*H*)

_{i}, H_{r+i}*= r,*there is no problem for

*r*3.

__>__ In case *r = *2, there is an essentially unique way to glue the 2 copies of in such a way that no new triangles are created, and this produces a triangulation of the orientable closed surface of genus 3 (the ''triple torus''). We verified this by computer using GAP [4]. It can be shown that the double torus does not admit a locally *C _{7}* triangulation.

Notice that for *r >* 3 the construction allows more freedom at the time of gluing (so in principle more than one example may have been constructed at each genus

*g >*3) and that even non-orientable surfaces are obtained gluing one handle in a non-orientable manner. So we have proved:

**Theorem 3.1 ***Every orientable surface of genus at least 3, and every non-orientable surface with even Euler characteristic *c* <* –6

*admits a locally C*(

_{7}triangulation G. For any such G, K*G*)

*is a positive answer to Prisner's open problem*1

*in*[11].

**References**

[1] Claude Berge. *Hypergraphs. *North-Holland Publishing Co., Amsterdam, 1989. Combinatorics of finite sets, Translated from the French. [ Links ]

[2] Morton Brown and Robert Connelly. On graphs with a constant link. In *New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), *pages 19-51. Academic Press, New York, 1973. [ Links ]

[3] C. H. Dowker. Homology groups of relations. *Ann. of Math. (2), *56:84-95, 1952. [ Links ]

[4] The GAP Group, Aachen, St Andrews. *GAP - Groups, Algorithms, and Programming, Version 4.2, *2000. (http://www-gap-system.org). [ Links ]

[5] Nora Hartsfield and Gerhard Ringel. Clean triangulations. *Combinatorica, *11 (2): 145-155, 1991. [ Links ]

[6] P. J. Hilton and S. Wylie. *Homology theory: An introduction to algebraic topology. *Cambridge University Press, New York, 1960. [ Links ]

[7] F. Larrión and V. Neumann-Lara. Locally *C§ *graphs are clique divergent. *Discrete Math., *215(1-3):159 170, 2000. [ Links ]

[8] F. Larrión, V. Neumann-Lara, and M. A. Pizaña. Whitney triangulations, local girth and iterated clique graphs. Discrete Math., 258(1-3):123-135, 2002. [ Links ]

[9] Erich Prisner. Homology of the line graph and of related graph-valued functions. *Arch. Math., *56(4):400-404, 1991. [ Links ]

[10] Erich Prisner. Convergence of iterated clique graphs. *Discrete Math., *103(2): 199-207, 1992. [ Links ]

[11] Erich Prisner. *Graph dynamics. *Longman, Harlow, 1995. [ Links ]

[12] C. P. Rourke and B. J. Sanderson. *Introduction to piecewise-linear topology. *Springer-Verlag, Berlin, 1982. Reprint. [ Links ]

[13] Edwin H. Spanier. *Algebraic topology. *Springer-Verlag, New York, 1981. Corrected reprint. [ Links ]

[14] Jayme L. Szwarcfiter. A survey on clique graphs. In Recent Advances in Algorithms and Combinatorics. C. Linhares and B. Reed, eds., Springer-Verlag. To appear. [ Links ]

[15] Jayme L. Szwarcfiter. Recognizing clique-Kelly graphs. *Ars Combin., *45:29-32, 1997. [ Links ]

[16] W. T. Tutte. A census of plane triangulations. *Cañad. J. Math., *(14):21-28, 1962. [ Links ]

[17] H. Whitney. A theorem on graphs. *Ann. Math., *32(2):378-390, 1931. [ Links ]

*Dedicated to Prof. J. L. Szwarcfiter in his 60* ^{th}* Anniversary.

^{**}Partially supported by CONACyT, Grant 400333-5-27968E.