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Journal of the Brazilian Computer Society

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

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



On the homotopy type of the clique graph*



F. LarriónI,1; V. Neumann-LaraI,1; M. A. PizañaII

IInstituto de Matemáticas, U.N.A.M., Circuito Exterior, C.U. México 04510 D.F. MÉXICO., {paco, neumann}
IIUniversidad Autónoma Metropolitana, Depto. de Ingeniería Eléctrica., Av. Michoacán y Purísima s/n México 09340 D.F. MÉXICO.,,




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 NG(x) induced in G by the (open) neighbourhood of any vertex a; Î G is isomorphic to H. We say that G is locally , = {H1,H2,...} if for every x Î G, NG(x) @ Hi for some Hi 6 . Cn and Pn are, respectively, the cyclic and path graphs on n vertices. We say that G is locally cyclic if it is locally {Cn :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 K0(G) = G and Kn+1(G) = K(Kn(G)). G is K-periodic if G @ Kn(G) for some n > 1. Extensive bibliography on clique graphs can be found in [14].

A graph G is clique-Helly if whenever X = {qi,...,qn } Í V(K(G)) is a family of pairwise intersecting cliques, then X ¹ Æ. We say that Q = {q1,...,qn } Î V(K2(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 K4) 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 {Cn,Pm : 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) K3(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 = {q1,q2,...,qr } 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 q0 Î K(G) is called a center of X if:

Y Í X and Y ¹ Æ imply (Y È{q0}) ¹ Æ .

Note that X È {q0 } 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 qi) plus a (formal) barycenter b(s) for each s Î , and whose simplexes are of the form {q1,...,qn , (s1),..., b(sm )} and satisfy:

1. {q1,...,qn}Î.

2. sj Î for all j.

3. si Î s1 for all i.

4. sj Í sj+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 K2 (G) : s Í Q}

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

Now define the map j1 : ' ® by j1 (qi) = qi and j1 (b(sj)) = q(j). Then for any simplex of ' we have that j1{q1,...,qn , (s1),..., b(sm)} = {q1,...,qn , (1),..., b()}. This is a simplex of because there is a clique Q of K(G) such that qi Î Q and sj Í Q for all i = l,...,n and j = 1,..., m (take a Q with sm Í Q). Therefore qi,q(j ) Î Q for all i and j. It follows that j1 : ' ® is a Simplicial map, so | j1| : |'| ® || is continuous.

We claim now that Im(j1 i) = : As q1 Ç ¼ Ç qn ¹ Æ and {q1,...,qn } Í s1 Í 1, we obtain that q1 Ç ¼ Ç qn q(1)¹ Æ. Using that 1 Í 2 Í ¼ Í and q(j ) Î j for all j, it follows by induction that {q1,...,qn , (1),..., b()} is a simplex of . Now we know that Im(|j1| ) = || and that the restriction of |j1|to || is the identity in ||.

On the other hand, consider the canonical homeomorphism j0 : || ® |'|. Let j = j1 o j0. 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 = {q1,...,qr } be a minimal subset of X also satisfying Z = Æ.

Since Z is minimal and necessarily r > 3, we may take x23 Î (Z — {q1}), x13 Î (Z — {q2}) and x12 Î (Z – {q3})- Hence, q0 = {x12, x13 , x23} 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 q1 = {x12, x13 , a}, q2 = {x12, x13 , b} and q3 = {x13 , x23, c} for some three (different) vertices a, b, c Î G. Since q1 Ç q2 Ç q3 = Æ it follows that Z = {q1 , q2 , q3 }.

Let Q Î K2(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 {x12, x13 , x23, a, b, c} induces an octahedron in G, contradicting our hypotheses. If |q Ç q0| = 1, say q n q0 = {x12}, then q Ç q3 = {c} and {x12, x13 , x23 , c} would contradict w(G) < 3. Therefore |q Ç q0| > 2 for every q Î Q.

Since the set {q Î K(G) : |q Ç q0| > 2} is a complete of K(G) it follows that Q = {q Î K(G) : |q Ç q0| > 2}. Now the condition on the clique number implies that q0 is a center of 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 {Ct : t > 7} graph 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 ´ K3 is a locally C10 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 Ct graph. Next, we shall construct an explicit infinite family of locally C7 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 C6 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 vT = (2r,4r). Let Gr be the translation group defined by u and vr, and let r be the parallelogram defined by these two vectors. The locally C6 graph Gr = T/Gr defines a Whitney triangulation of the torus with 14r vertices: Gr is obtained by identifying the parallel edges of r.

Now consider the 2r vertices w1, w2, w2r of Gr which correspond to the vertices (2,1),(3,3),...,(2r + 1,4r - 1) in r, i.e. wi = (i + 1,2i — 1). The vertices of GT are the disjoint union of the closed neighbourhoods N[wi] of these vertices, and removing these vertices from Gr we obtain a locally P5 graph of order 12r. Let us call r the surface triangulated by , which is a torus with 2r open disks removed. All the vertices of lie in the border of r. The connected components of the border of r are the hexagons H1, H2,...,H2r which were the open neighbourhoods of the removed vertices w1, w2,..., w2r of Gr.

Consider the locally P4 graph in figure 3. This graph gives us a Whitney triangulation of a cylinder, all the vertices lie in the border whose connected components B1 and B2 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 B1 with HI and B2 with Hr+1in an orientable manner, so a handle is glued to Sr. For the second copy, identify B1 with H1 and B2 with Hr+1, so a second handle is glued tor. Continuing in this way, we obtain at the end a closed surface which is a sphere with r + 1 handles. The graph r obtained from and the 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 C7 we have to take care so that the triangles in r are exactly the triangles already present (16r inand 12 in each copy of ). This fails when two vertices x Î Hi and y Î Hi+r with d(x,y) < 3 in are identified with adjacent vertices in the i-th copy of . Since d(Hi, Hr+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 C7 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 C7 triangulation G. For any such G, K(G) is a positive answer to Prisner's open problem 1 in [11].



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*Dedicated to Prof. J. L. Szwarcfiter in his 60th Anniversary.
**Partially supported by CONACyT, Grant 400333-5-27968E.

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