On the homotopy type of the clique graph

Abstract

If G is a graph, its clique graph K(G) is the intersection graph of all its (maximal) cliques. The complex G<FONT FACE=Symbol></FONT> 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<FONT FACE=Symbol></FONT> and K(G)<FONT FACE=Symbol></FONT> 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?

clique graphs; clique convergence; Whitney triangulations; clean triangulations; simplicial complexes; modulo 2 Betti numbers

ARTICLES

On the homotopy type of the clique graph^*

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

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

^IIUniversidad Autónoma Metropolitana, Depto. de Ingeniería Eléctrica., Av. Michoacán y Purísima s/n México 09340 D.F. MÉXICO., [e-mail], 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 , = {H₁,H₂,...} if for every x Î G, N_G(x) @ H_i for some H_i 6 . C_n and P_n are, respectively, the cyclic and path graphs on 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 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 = {q_i,...,q_n } Í V(K(G)) is a family of pairwise intersecting cliques, then X ¹ Æ. We say that Q = {q₁,...,q_n } 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 {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) 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,...,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 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 q_i) plus a (formal) barycenter b(s) for each s Î – , and whose simplexes are of the form {q₁,...,q_n , (s₁),..., b(s_m )} and satisfy:

1. {q₁,...,q_n}Î.

2. s_jÎ – for all j.

3. s_iÎ s₁ for all i.

4. s_jÍ 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 K² (G) : s Í Q}

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

Now define the map j₁ : ' ® by j₁ (q_i) = q_i and j₁ (b(s_j)) = q(_j). Then for any simplex of ' we have that j₁{q₁,...,q_n , (s₁),..., b(s_m)} = {q₁,...,q_n, (₁),..., b()}. This is a simplex of because there is a clique Q of K(G) such that q_i Î Q and s_jÍ Q for all i = l,...,n and j = 1,..., m (take a Q with s_mÍ Q). Therefore q_i,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 q₁Ç ¼ Ç q_n¹ Æ and {q₁,...,q_n} Í s₁Í ₁, we obtain that q₁Ç ¼ Ç q_n 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,...,q_n , (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,...,q_r} be a minimal subset of X also satisfying Z = Æ.

Since Z is minimal and necessarily r > 3, we may take x₂₃Î (Z — {q₁}), x₁₃Î (Z — {q₂}) and x₁₂Î (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 {C_t : 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 C_tgraph. 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 C₆graph 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_T = (2r,4r). Let G_r be the translation group defined by u and v_r, and let _rbe the parallelogram defined by these two vectors. The locally C₆graph G_r = T/G_r defines a Whitney triangulation of the torus with 14r vertices: G_ris obtained by identifying the parallel edges of _r.

Now consider the 2r vertices w₁, w₂, w_2r of G_rwhich correspond to the vertices (2,1),(3,3),...,(2r + 1,4r - 1) in _r, i.e. w_i = (i + 1,2i — 1). The vertices of G_Tare the disjoint union of the closed neighbourhoods N[w_i] of these vertices, and removing these vertices from G_rwe obtain a locally P₅ graph of order 12r. Let us call _rthe 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 _rare the hexagons H₁, H₂,...,H_2r which were the open neighbourhoods of the removed vertices w₁, w₂,..., w_2rof G_r.

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 B₂with H_r+1in an orientable manner, so a handle is glued to S_r. For the second copy, identify B₁ with H₁and B₂with H_r+1, so a second handle is glued to_r. 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

_rto be locally C₇ we have to take care so that the triangles in _rare exactly the triangles already present (16r inand 12 in each copy of ). This fails when two vertices x Î H_i and y Î H_i+rwith 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 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 60

^th Anniversary.

^**

Partially supported by CONACyT, Grant 400333-5-27968E.

Publication Dates

• Publication in this collection
  16 Dec 2003
• Date of issue
  2001