The theory of complexes of directed trees was initiated by Kozlov to answer a question by Stanley, and later on, results from the theory were used by Babson and Kozlov in their proof of the Lovasz conjecture. We develop the theory and prove that complexes on directed acyclic graphs are shellable.
A related concept is that of independence complexes: construct a simplicial complex on the vertex set of a graph, by including each independent set of vertices as a simplex. Two theorems used for breaking and gluing such complexes are proved and applied to generalize results by Kozlov.
A fruitful restriction is anti-Rips complexes: a subset P of a metric space is the vertex set of the complex, and we include as a simplex each subset of P with no pair of points within distance r. For any finite subset P of \mathbb{R} the homotopy type of the anti-Rips complex is determined.
We use two cofibre sequences to identify some combinatorial situations when the independence complex of a graph splits into a wedge sum of smaller independence complexes. Our main application is to give a recursive relation for the homotopy types of the independence complexes of powers of cycles, which answers an open question of D. Kozlov.
We determine the homotopy types of the independence complexes of the $(n \times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.
We prove that the independence complexes of some grids have exponential Betti numbers. This corresponds to the number of ground states in the hard-core model in statistical physics with fermions in the vertices of the grid.
We show that if a graph $G$ involves a certain square grid graph as a full subgraph, then a certain operation on it yields a simplicial suspension of the independence complex of $G$. This generalizes a result of Csorba. As a corollary, we determine the simple homotopy types of the independence complexes of some grid graphs.
We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.
We introduce the notion of \textit{star cluster} of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from works of Engström and Jonsson on the homotopy type of independence complexes of triangle-free graphs and to investigate a large number of examples which appear in the literature. We present an alternative way to study the chromatic number of a graph from a homotopical point of view and obtain new results regarding the connectivity of independence complexes.
In this article, we introduce the notion of a wedge of graphs and provide detailed computations for the independence complex of a wedge of path and cycle graphs. In particular, we show that these complexes are either contractible or wedges of spheres.
The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(Δ_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(Δ_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(Δ_n^m)$ and prove that several homology groups of $\mathrm{Ind}(Δ_n^m)$ are non-zero.
The independence complex of a graph G is a simplicial complex whose simplices are the independent sets in G. In the last couple of decades, the independence complexes of square grids (with various boundary conditions) have gained much attention because of their connections with the hard square model from statistical physics. In this article, we prove that if G is an $m\times n$ grid with open or cylindrical boundary condition then its independence complex is homotopy equivalent to a wedge of spheres. A part of this result settles a conjecture of Iriye.
For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.
The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction induces a spectral sequence that converges to zero and contains on its first page the homology of the independence complexes of $G$ and various subgraphs of $G$, obtained by removing independent sets and their neighborhoods from $G$. It is shown that this spectral sequence may be u…
For integers n\geq 1, k\geq 0, the stable Kneser graph SG_{n,k} (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n+k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form {i,i+1} or {1,2n+k}. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number k+2. This article contains a study of the independence complexes of SG_{n,k} for small values of n and k. Our contributions a…
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the introduced notions.
For $r\geq 1$, the $r$-independence complex of a graph $G$, denoted Ind$_r(G)$, is a simplicial complex whose faces are subsets $A \subseteq V(G)$ such that each component of the induced subgraph $G[A]$ has at most $r$ vertices. In this article, we establish a relation between the distance $r$-domination number of $G$ and (homological) connectivity of Ind$_r(G)$. We also prove that Ind$_r(G)$, for a chordal graph $G$, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose $r$-independence complex ha…
In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions.
We present a number of lower bounds for the h-vectors of k-CM, broken circuit and independence complexes. These lead to bounds on the coefficients of the characteristic and reliability polynomials of matroids. The main techniques are the use of series and parallel constructions on matroids and the short simplicial h-vector for pure complexes.
For each finite simple graph $G$, Aharoni, Berger and Ziv consider a recursively defined number $ψ(G) \in \mathbb{Z}\cup \{+ \infty \}$ which gives a lower bound for the topological connectivity of the independence complex $I_G$. They conjecture that this bound is optimal for every graph. We use a result of recursion theory to give a short disproof of this claim.
We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join $\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedge sum of CW complexes of the form $Σ^r X_{i_1} * X_{i_2} * \cdots * X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CW complexes and $Σ$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over …
The bounded degree complex $BD(G,λ)$ is a generalization of the matching complexes of a graph. In this paper, we show that the bounded degree complex of a forest is shellable, by using independence complexes of hypergraphs. We obtain a wedge decomposition result of bounded degree complexes when a graph $G$ has a leaf
We study the class of independence complexes of claw-free graphs. The main theorem give good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is (cn/d+epsilon)-connected, where c=2/3. This can be compared with the result of Szabo and Tardos that c=1/2 is optimal with no restrictions on the graphs. Secondly, we calculate the connectivity of a family of complexes used in Babson and Kozlov's proof of Lovasz co…
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.
We study the independence complexes of families of well-covered circulant graphs discovered by Boros-Gurvich-Milanič, Brown-Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g. vertex decomposable, shellable) or topological (e.g. Cohen-Macaulay, Buchsbaum) structure. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen-Macaulay, and/or Buchsbaum. …
Shelling orders are a ubiquitous tool used to understand invariants of cell complexes. Significant effort has been made to develop techniques to decide when a given complex is shellable. However, empirical evidence shows that some shelling orders are better than others. In this article, we explore this phenomenon in the case of matroid independence complexes. Based on a new relation between shellability of dual matroid polytopes and independence complexes, we outline a systematic way to investigate and compare different shellings orders. We explain how our new tools recast and deepen various c…
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$. We study the reduced Euler characteristic $\tildeχ$ of the independence complex $Δ(G)$ for $n=p^k$ with $p$ prime and for $n=2p^k$ with $p$ odd prime, proving that in both cases $\tildeχ$ does not vanish. We also give an example of circulant graph whose independence complex has $\tildeχ$ equals to $0$, giving a negative answer to R. Hoshino.
We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.
Faculty of Education, An Giang University Vietnam National University Ho Chi Minh City（安江大学越南国立大学胡志明市教育学院）；Institute of Mathematics, VAST（越南科学技术院数学研究所）
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.
This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests on a double directed string, while the other one consists of Shapiro-Welker models for the spaces of hyperbolic polynomials with a triple root. We explain this coincidence in the more general context by finding an explicit homotopy equivalence between complexes of directed forests on a double directed tree, and doubly disconnecting complexes of a tree.
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated generating function, of the multipath complex for some families of graphs, including transitive tournaments and complete bipartite graphs. Then, we compute the homotopy type of multipath complexes of linear graphs, polygons, small grids and transitive tournaments. We show that they are all contractible or wedges of spheres. We introduce a new technique for deco…
The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$. Braun and Hough introduced a family of graphs $Δ^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $Δ^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.
The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the crosspolytopes allow such a shelling. Furthermore, we show that the complex of directed trees of a complete double directed graph is a union of suitable spheres. We also investigate shellability of the maximal pure skeleton of a complex of directed trees.…
We prove that the inclusion from oriented graph complex into graph complex with at least one source is a quasi-isomorphism, showing that homology of the "sourced" graph complex is also equal to the homology of standard Kontsevich's graph complex. This result may have applications in theory of multi-vector fields $T_{\rm poly}^{\geq 1}$ of degree at least one, and to the hairy graph complex which computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play …
We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. We give some applications of this method to complexes arising from graphs. As a consequence, we answer some questions raised in [R.Ehrenborg, G.Hetyei, European J.Combin.27 (2006), no.6, 906-923] on the independence complex and the dominance complex. The techniques used come mainly from monomial ideal theory.
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the (lower-)associated independence hypergraph of the random hypergraph in terms of the probability distributions of certain random simplicial complex and certain random independence hypergraph of Erdös-Rényi type. We construct a graded structure of the map algebra explicitly and give algorithms to generate random hypergraphs and random simplicial complexes.
In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the path homology and cohomology of a digraph.
We introduce WI-posets as intermediate objects in the study of Z_2-homotopy types of graph complexes. It turns out that (almost) all graph complexes associated to a graph can be viewed as avatars of the same object, as long as their Z_2-homotopy types are concerned. Among the applications are a proof that each finite, free Z_2-complex is a graph complex and an evaluation of Z_2-homotopy types of complexes Ind(C_n) of independence sets in a cycle C_n. The main tools used in the paper are Quillen fiber theorem and Bredon criterion for Z_2-equivalence of Z_2-complexes.
We analyze the quantum ground state structure of a specific model of itinerant, strongly interacting lattice fermions. The interactions are tuned to make the model supersymmetric. Due to this, quantum ground states are in one-to-one correspondence with cohomology classes of the so-called independence complex of the lattice. Our main result is a complete description of the cohomology, and thereby of the quantum ground states, for a two-dimensional square lattice with periodic boundary conditions. Our work builds on results by J. Jonsson, who determined the Euler characteristic (Witten index) vi…
