A note on independence complexes of chordal graphs and dismantling

We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.


Introduction
Chordal graphs, that is graphs without an induced cycle of length more than three, form one of the fundamental classes in graph theory. They have a well known characterization in the realm of combinatorial topology: Fact. Let G be a simple, undirected graph. Then the following are equivalent: (a) G is chordal.
The clique complex of every connected induced subgraph of G is contractible.
(c) The clique complex of every connected induced subgraph of G is dismantlable.
(d) Every connected induced subgraph of G is dismantlable.
The purpose of this note is to show a related result in the dual case of independence complexes of chordal graphs. Theorem 1. Suppose G is a chordal graph. Then the independence complex of G is contractible if and only if it is dismantlable, or equivalently if G is dismantlable as a graph.
In the last section we extend this result to other homotopy types of the independence complex (Theorem 9).
Independence complexes of chordal graphs have received due attention in the literature [10,9,8,15,20,22,6]. They are vertex-decomposable [22], hence homotopy equivalent to wedges of spheres or to a point. Every finite wedges sum of spheres is realizable as the homotopy type of the independence complex of some chordal graph [15]. Moreover, every homology class in the independence complex of a chordal graph is represented by a crosspolytopal sphere corresponding to some induced matching in the graph [2] (called crosscycle in [14]). The strong connection between topology and combinatorics for independence complexes of chordal graphs makes it plausible that also contractibility of these spaces has a combinatorial manifestation.
Dismantling is a simple operation which, if available, reduces the size of a simplicial complex without changing its homotopy type. A single dismantling step consists of removing a vertex whose link is a cone. This operation has been studied independently by many authors under the names strong collapse [3], LC-reduction [5] or link-cone reduction [18]. A sequence of dismantling steps reduces a simplicial complex to a core subcomplex whose isomorphism type does not depend on the choices made in the process [3,18,13]. A complex is dismantlable if its core is a single vertex. There is a compatible notion of dismantlability for graphs [13], so that the clique complex of G is dismantlable if and only if G is dismantlable as a graph. Dismantlable graphs are also known as cop-win graphs [19].
There are other combinatorial reduction schemes of this kind: non-evasiveness and collapsibility [16]. A complex is non-evasive if it is a single point or if it has a vertex whose link and deletion are both non-evasive. A complex is collapsible if it can be reduced to a single vertex by removing free faces. Dismantlability implies non-evasiveness, which implies collapsibility, and that in turn implies contractibility, but neither implication can be reversed. It is easy to observe (Lemma 5.(d)) that a contractible independence complex of a chordal graph is non-evasive. It is the upgrade of this statement from non-evasive to dismantlable that requires more work. In general, dismantlability of simplicial complexes does not follow from the combination (vertex-decomposable + contractible).
On the practical side, the advantage of dismantlability over non-evasiveness or collapsibility is that it is very easy to check. As mentioned before, dismantlability can be tested greedily, by removing any available vertex whose link is a cone and continuing in the same fashion with the smaller complex. In our case this easily leads to an algorithm which solves the decision problem "Is the independence complex of a given chordal graph G contractible?" in time O(n 3 ), where n is the number of vertices in G. If the complex is contractible, the algorithm provides a dismantling sequence. It may be compared with the algorithm of [2], which solves the same problem in time O(m 2 ), where m is the number of edges. On the other hand, if the complex is not contractible, the algorithm of [2] additionally computes an induced matching representing some non-zero homology class.

Prerequisites
For a simplicial complex K the link and deletion of a vertex v will be denoted lk K (v) and K \ v, respectively. If G is a simple, undirected graph then the clique complex Cl(G) and the independence complex Ind(G) are the simplicial complexes on the vertex set of G, whose faces are all the cliques, resp. all the independent sets of G. Clearly Ind(G) = Cl(G), where G is the complement of G. The subgraph of G induced by a vertex set The complete graph on n vertices is denoted K n . We write N G (u) = {w : uw ∈ E(G)} for the open neighbourhood and N G [u] = N G (u) ∪ {u} for the closed neighbourhood of a vertex u in G. Then we have A vertex u of a simplicial complex K is said to be dominated by a vertex u if the link lk K (u) is a cone with apex u . The removal of a dominated vertex u is called an elementary dismantling. A complex K is dismantlable if there is a sequence of elementary dismantlings from K to a point. In the language of [3] K is dismantlable if it has the strong homotopy type of a point.
We will now discuss the same concepts for graphs, see [13,Sect. 2.11] . If G can be reduced to a single vertex by successive removals of dominated vertices, then we say G is dismantlable (or cop-win). 1 Since holds if and only if lk Cl(G) (u) is a cone with apex u , we immediately get the next observation.
Let us make things more explicit for independence complexes.
, and the latter is equivalent to N G (u ) ⊆ N G (u). We will give this case a name.
Note that if (x, y) is a good pair in G then xy ∈ E(G). As discussed before, if (x, y) is a good pair in G then y is dominated by x in G and in Ind(G) and the removal of y is an elementary dismantling of Ind(G). 2 We can now state an equivalent version of Theorem 1.

Theorem 4.
If G is a chordal graph such that Ind(G) is contractible then either G is a single vertex or G has a good pair.
This statement immediately implies Theorem 1 since the class of chordal graphs is hereditary, that is closed for induced subgraphs. The proof of Theorem 4 occupies the next section. In the remaining part of this section we recall various elementary properties of chordal graphs. A For the purpose of this note a vertex u of G will be called peeling if some neighbour of u is simplicial. 3 It is a fundamental theorem of Dirac that every chordal graph has a simplicial vertex [7,21]. It follows that a chordal graph with at least one edge has a peeling vertex.
The next lemma is a compilation of various topological prerequisites related to independence complexes of chordal graphs. It also contains the weak analogue of Theorem 1 with dismantlable replaced by non-evasive.
(a) [1,10,15] If u is a peeling vertex then there is a homotopy equivalence [22,20,8,15] Ind(G) is homotopy equivalent to a wedge of spheres or it is contractible.
1 It should be pointed out that some authors call u dominated by u when NG(u) ⊆ NG(u ). This gives a different notion of dismantling, which need not preserve the homotopy type of Cl(G).
2 This is not the same as the notions of co-domination and co-dismantling defined in [4], since they are based on the condition NG[x] ⊆ NG[y], and need not preserve the homotopy type of Ind(G). 3 Every peeling vertex of G is a shedding vertex of Ind(G) in the sense of vertex-decomposability. See [22]. Finally, for (f), note that if two vertices z, z ∈ N G (x) were not adjacent, then x, z, y, z would be an induced 4-cycle in G since xy ∈ E(G).

The main theorem
In contrast with the simple argument of Lemma 5.(d), where any peeling vertex u does the job, not every simplicial vertex x is part of a good pair (x, y). That makes the proof of Theorem 4 more complicated. To locate good pairs we will employ the characterization of chordal graphs as intersection graphs of subtrees of a tree (see [21]).
We say that a tree T (whose vertices will be called nodes) is a tree model of a graph G if: (a) every node of T contains a subset of vertices of G (perhaps empty), (b) for every vertex v ∈ V (G), the nodes of T which contain v form a subtree of T , (c) vw ∈ E(G) if and only if there is a node of T containing both v and w. A node of T which contains a vertex v will be called a v-node. Now (b) states that for each vertex v, the v-nodes span a tree. A graph is chordal if and only if it has a tree model [11]. One common construction is a clique tree, which has the additional property that the nodes of T are in bijection with maximal cliques of G, but we do not make any such extra restrictions. If R is any marked node of T (the root), then we call (T, R) a rooted tree model of G.
If T is a tree model of G and W ⊆ V (G) is any subset, then T | W will be the tree obtained from T by retaining only the vertex labels that belong to W and erasing those not in W , without changing the shape of T . Then For the inductive arguments we will need good pairs which are compatible with a given rooted tree model of G in the sense defined next. Definition 6. Suppose (T, R) is a rooted tree model of G. We say that (x, y) is (T, R)-good in G if (x, y) is good in G and the shortest path in T from any x-node to R contains some y-node.
Intuitively, a pair (x, y) is (T, R)-good if the x-nodes are located deeper in (T, R) than the y-nodes. The next lemma is used to promote good pairs from certain subgraphs of G to good pairs in G. R) is any rooted tree model of G and G is a connected component of G.
If (x, y) is a (T | V (G ) , R)-good pair in G then it is a (T, R)-good pair in G. In particular, in either case, x is a simplicial vertex of G by Lemma 5.(f ).
Proof. Part (a) is obvious since the assumptions imply N G (x) ⊆ N G [u] and xu ∈ E(G).
For part (b), consider any vertex z with xz ∈ E(G). We want to show that yz ∈ E(G).
. Then there must exist two nodes U, X in T such that U is a u-node, X is an x-node and both X, U are z-nodes. Since the set of z-nodes forms a subtree containing X and U , every node on the shortest path P from X to U in T is also a z-node. The path P induces a path P from X to in T /u. The path P contains a y-node, by the assumption that (x, y) is (T /u, )-good. It follows that P contains a y-node, and therefore yz ∈ E(G).
Part (c) is clear since the vertices from the other components of G do not affect the neighbourhoods of x nor y.
Next comes the key part of the argument, where we inductively construct good pairs with respect to arbitrary rooted tree models in connected chordal graphs. Proof. We prove the theorem by induction on the number of vertices of G. It is obviously true when G has at most one vertex. Now suppose G is any connected chordal graph with at least two vertices for which Ind(G) is contractible. Fix an arbitrary rooted tree model (T, R) of G. For any vertex v in G let f (v) denote the minimal distance in T from any v-node to R. Since G is chordal and connected, it has at least one peeling vertex. Pick u to be any peeling vertex which minimizes f (u ) among all peeling vertices u of G. Case 1. G has more than one vertex. Consider the graph G together with a rooted tree model By induction there is a pair (x, y) in G which is good with respect to this model. The same pair is (T /u, )-good in G \ N G [u] by Lemma 7.(c). Now Lemma 7.(b) implies that the pair (x, y) is good in G. It remains to show that it is (T, R)-good. Let X and U be the x-node and the u-node which minimize the distance in T from any x-node to any u-node. Since xu ∈ E(G), these nodes are well and uniquely defined. The assumption that (x, y) is (T /u, )-good in G \ N G [u] implies that the shortest path in T from X to U contains a y-node. Let Y be the y-node on that path which is closest to X. Then X and Y also minimize the distance from any x-node to any y-node. See Figure 3.
Since G is connected, and therefore N G (x) = ∅, there exists a vertex u ∈ V (G ) ⊆ V (G) such that xu , yu ∈ E(G ) ⊆ E(G). The set of u -nodes in T spans a subtree, and it follows that all nodes on the shortest path from X to Y (inclusive) are u -nodes. Note that u is peeling in G since its neighbour x is simplicial in G by Lemma 5.(f).
Consider all possible locations of R in T . If R and X lie in the same component of T \ Y then f (u ) < f (u), contradicting our choice of u. It follows that either R = Y or R and X lie in different components of T \ Y . In these cases the shortest path from any x-node to R visits Y , proving that (x, y) is (T, R)-good in G.
Case 2. G is a single vertex x. By Lemma 7.(a) the pair (x, u) is good in G. As before, let X and U be the x-node and the u-node which minimize the distance in T from any x-node to any u-node. Since G is connected, there exists a vertex u ∈ V (G) such that xu , uu ∈ E(G), and therefore each node on the shortest path from X to U is a u -node. Since x is simplicial in G, its neighbour u is peeling in G. Now if R and X are in the same component of T \ U then f (u ) < f (u) and we have a contradiction. Hence R = U or R and U are in different components of T \ U , and then every path from an x-node to R visits U , showing that (x, u) is (T, R)-good in G.
It remains to prove Theorem 4 (and thus Theorem 1).
Proof of Theorem 4. Let G be a chordal graph with contractible Ind(G) and with at least two vertices. If G is connected, then it has a good pair by Theorem 8. If some connected component of G is a single vertex x, then (x, y) is a good pair in G for any other vertex y. Finally, if some connected component G of G, with more than one vertex, satisfies that Ind(G ) is contractible, then the pair (x, y) which is good in G is also good in G. By Lemma 5.(e) that covers all the possibilities.

Other results about cores
Even if Ind(G) is not contractible, one can still apply elementary dismantlings to reduce the size of the complex. The process stops when we reach a complex without a dominated vertex (such complexes are called taut in [12] and minimal in [3]), which we call the core of the original complex. It is well-defined up to isomorphism regardless of the order of elementary dismantlings, see [3,Thm. 2.11], [18] for complexes and [13, Thm. 2.60] for graphs. Of course Ind(G) is taut if and only if G has no good pair.
For example, let F be a forest. It is easy to see that a forest without a good pair is either a single vertex or a disjoint union of edges. It follows that the core of Ind(F ) is either a point or the boundary complex of some cross-polytope. The fact that Ind(F ) is homotopy equivalent to a point or a sphere was first proved in [9], and the statement about cross-polytopal cores is implicitly contained in [17].
The next result extends Theorem 1 and generalizes the example from the previous paragraph, by considering the case when G is an arbitrary chordal graph and Ind(G) is homotopy equivalent to a single sphere (i.e. has total reduced Betti number 1). Let M k denote the matching with k edges, that is the graph with 2k vertices whose each connected component is an edge. The independence complex Ind(M k ) is isomorphic to the boundary complex of the k-dimensional cross-polytope.
Theorem 9. If G is a chordal graph with a homotopy equivalence Ind(G) S k−1 for some k ≥ 1 then the core of Ind(G) is isomorphic to Ind(M k ).
Proof. It suffices to prove the following claim: If G is a chordal graph without a good pair and Ind(G) S k−1 then G is isomorphic to M k . Take any graph G as in the claim and fix any tree model T of G.
We will first show that for any vertex u of G the complex Ind(G\N G [u]) is not contractible. Suppose otherwise, and let G be some connected component of G \ N G [u] such that Ind(G ) is contractible (it exists by Lemma 5.(e)). If G is a single vertex x then (x, u) is a good pair in G by Lemma 7.(a), a contradiction. If G has more than one vertex then, by Theorem 8, there is a pair (x, y) which is ((T /u)| V (G ) , )-good in G . By Lemma 7 parts (c) and (b) this pair is also good in G, which is again a contradiction. It shows that Ind(G \ N G [u]) is not contractible.
An inductive application of Lemma 5.(a) shows that if v is a simplicial vertex of G then there is a homotopy equivalence see also [10,Thm. 3.5]. If v has degree at least 2 then the wedge sum in (3) has at least 2 summands, and by the previous observation each of them is homotopy equivalent to a noncontractible wedge of spheres. That contradicts the fact that Ind(G) is homotopy equivalent to a single sphere. Consequently, every simplicial vertex of G has degree 0 or 1. The first option is impossible, since then Ind(G) would be contractible. It follows that every simplicial vertex of G has degree 1.
If v is a vertex of degree 1 and y is any vertex in distance 2 from v then the pair (v, y) is good in G. It means that every simplicial vertex of G has no vertices at distance 2, and it easily follows that every connected component of G is a single edge, i.e. G is a matching. A comparison of dimensions shows that this matching must be M k , which ends the proof.
This yields a polynomial-time algorithm for the problem of checking if Ind(G) is homotopy equivalent to a sphere for a chordal graph G.
For wedges of spheres with higher Betti numbers the situation is more complicated. For example, there are (at least) two chordal graphs whose independence complex is taut and homotopy equivalent to S 1 ∨ S 1 . The first one is the disjoint union of K 3 and K 2 and the other is the 7-vertex graph obtained from three copies abc, def, ghi of K 3 by identifying c with d and f with g. The best we can offer is the following.
Conjecture 10. For every finite wedge sum of spheres X there are only finitely many chordal graphs G such that G has no good pair (i.e. Ind(G) is taut) and Ind(G) X.
The results of this note imply the conjecture when X is a point or a single sphere. We leave it as an exercise to check that it holds also when X is a wedge sum of copies of S 0 .