Homotopy theory of graphs

Babson, Eric; Barcelo, Hélène; Longueville, Mark de; Laubenbacher, Reinhard

doi:10.1007/s10801-006-9100-0

Cited by 61 publications

(141 citation statements)

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“…In other words χ(K) is the minimum number m such that there exists a nondegenerate simplicial map f : (1) ) is the vertex-edge graph of the complex K. In particular χ(K) reduces to the usual chromatic number if K is a graph, that is if K is a onedimensional simplicial complex.…”

Section: Chromatic Number χ(K) and Its Relativesmentioning

confidence: 99%

“…A simplicial complex is a flag-complex if σ ∈ K if and only if the associated 1-skeleton σ (1) of σ is a subcomplex of K. Examples of flag-complexes include the clique-complex Clique(G) of a graph and the order complex (P ) of all chains (flags) in a poset P . If K is a flag-complex and G = G K := K (1) is the associated vertex-edge graph of K, then K = Clique(G).…”

Section: Flag Complexes and The Homotopy Test Graphsmentioning

confidence: 99%

“…If K is a flag-complex and G = G K := K (1) is the associated vertex-edge graph of K, then K = Clique(G). From here we deduce, relying on the isomorphism given in Example 4.8, that if K and L are flag complexes, then…”

Section: Flag Complexes and The Homotopy Test Graphsmentioning

confidence: 99%

“…5. Moreover, for all associated graphs χ(W (1) ) = 4 which in light of (19) means that they are all candidates for "homotopy test graphs" in the sense of [34, Chap. 6].…”

Section: Hom(t G) Is K-connected ⇒ χ(G)mentioning

confidence: 99%

“…Combinatorial groupoids have appeared (implicitly) in other contemporary combinatorial constructions and applications, see Babson et al [1], recent work of Barcelo et al on Atkin-homotopy [5,6] etc. Bolker, Guillemin, and Holm [10] defined connections on graphs and, guided by the analogies between graphs and manifolds, applied them to the parallel redrawing problem.…”

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confidence: 99%

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Combinatorial Groupoids, Cubical Complexes, and the Lovász Conjecture

Živaljević

2008

Discrete Comput Geom

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This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature", etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial "Theorema Egregium" for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend BabsonKozlov-Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs.Keywords Combinatorial groupoids · Lovász conjecture · Cubical complexes · Discrete differential geometry IntroductionThere has been new and encouraging evidence that the language and methods of the theory of groupoids, after being successfully applied in other major mathematical fields, offer new insights and perspectives for applications in combinatorics and discrete and computational geometry.The groupoids (groups of projectivities) have recently appeared in geometric combinatorics in the work of M. Joswig [28]; see also a related paper with Izmestiev [27] and the references to these papers, where they have been applied to toric manifolds, branched coverings over S 3 , colorings of simple polytopes etc.

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Section: Chromatic Number χ(K) and Its Relativesmentioning

confidence: 99%

Section: Flag Complexes and The Homotopy Test Graphsmentioning

confidence: 99%

Section: Flag Complexes and The Homotopy Test Graphsmentioning

confidence: 99%

“…5. Moreover, for all associated graphs χ(W (1) ) = 4 which in light of (19) means that they are all candidates for "homotopy test graphs" in the sense of [34, Chap. 6].…”

Section: Hom(t G) Is K-connected ⇒ χ(G)mentioning

confidence: 99%

mentioning

confidence: 99%

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Combinatorial Groupoids, Cubical Complexes, and the Lovász Conjecture

Živaljević

2008

Discrete Comput Geom

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Early Examples of Software in Mathematical Knowledge Management

Ion

2014

Mathematical Software – ICMS 2014

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Homological Algebra for Persistence Modules

Bubenik

Milićević

2021

Found Comput Math

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We useČech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain six homology and homotopy theories of closure spaces. 1 Closure spacesIn this section we provide background on EduardČech's closure spaces [9,25].2.1. Elementary definitions. We start with the elementary definitions we need to work with closure spaces. Definition 2.1. Let X be a set and let P(X) denote the collection of subsets of X. A function c : P(X) → P(X) is called a closure operation (or just closure) for X if the following axioms are satisfied:for all A, B ⊆ X. An ordered pair (X, c) where X is a set and c a closure for X is called a (Čech) closure space. Elements of X are called points.Lemma 2.2. Let (X, c) be a closure space. If A ⊆ B are subsets of X then c(A) ⊆ c(B). Proof. Note thatExample 2.3. Let X be a set. Then the identity map 1 P(X) : P(X) → P(X) is a closure operation for X. It is called the discrete closure for X. The closure operation defined by the map A → X for A = ∅ and ∅ → ∅ is called the indiscrete closure for X.Definition 2.5. Given a set X and a closure c for X we can associate to it an interior operation for X, int c : P(X) → P(X) given by int c (A) := X − c(X − A).The interior operation satisfies the following:1) int c (X) = X, 2) For all A ⊆ X, int c (A) ⊆ A, 3) For all A, B ⊆ X, int c (A ∩ B) = int c (A) ∩ int c (B). If int : P(X) → P(X) is a function satisfying the 3 conditions in Definition 2.5 and one defines c int : P(X) → P(X) by c int (A) := X − int(X − A), then c int is a closure operation for X.Proposition 2.6. [9, 14.A.12] A subset A of a closure space (X, c) is open if and only if int c (A) = A.Definition 2.7. [9, Definition 14.B.1] Let (X, c) be a closure space. Let A ⊆ X. We say B ⊆ X is a neighborhood of A if A ⊆ X − c(X − B) = int c (B). In the case that A = {x}, we say B is a neighborhood of x. The neighborhood system of A is the collection of all neighborhoods of A. Definition 2.8. [9, Definition 16.A.1] Let (X, c X ) and (Y, c Y ) be closure spaces. A map). If f is continuous at every point of X we say f is continuous. Equivalently, f is continuous if for every A ⊆ X, f (c X (A)) ⊆ c Y (f (A)). A continuous map f is called a homeomorphism if f is a bijection with a continuous inverse.

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Homotopy theory of graphs

Cited by 61 publications

References 7 publications

Combinatorial Groupoids, Cubical Complexes, and the Lovász Conjecture

Combinatorial Groupoids, Cubical Complexes, and the Lovász Conjecture

Early Examples of Software in Mathematical Knowledge Management

Homological Algebra for Persistence Modules

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