Gromov hyperbolic graphs

Bermudo, Sergio; Rodrı́guez, José M.; Sigarreta, José M.; Vilaire, Jean‐Marie

doi:10.1016/j.disc.2013.04.009

Cited by 61 publications

(60 citation statements)

References 23 publications

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“…Graph hyperbolicity provides tight bounds on the worst additive distortion of the distances in a (connected) graph when its vertices are embedded into a weighted tree. Several definitions exist, some of them considering graph metrics that are slightly different from the usual shortest-path metric, but they are equivalent to it up to a linear-function [5,17,26]. Moreover, 0-hyperbolic graphs are exactly the connected graphs whose given metric is a tree metric, which makes hyperbolicity a tree-likeness parameter.…”

Section: Definitions and Notationsmentioning

confidence: 99%

Applying clique-decomposition for computing Gromov hyperbolicity

Cohen

Coudert

Ducoffe

et al. 2017

Theoretical Computer Science

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The shortest-path metric d of a connected graph G is δ-hyperbolic if, and only if, it satisfies d (u, v)+d(x, y) ≤ max{d(u, x)+d(v, y), d(u, y)+d(v, x)}+2δ, for every 4-tuple u, x, v, y of G. We investigate some relations between the hyperbolicity of a graph and the hyperbolicity of its atoms, that are the subgraphs resulting from the clique-decomposition invented by Tarjan [34,45]. More precisely, we prove that the maximum hyperbolicity taken over all the atoms is at least the hyperbolicity of G minus one. We also give an algorithm to slightly modify the atoms, which is at no extra cost than computing the atoms themselves, and so that the maximum hyperbolicity taken over all the resulting graphs is exactly the hyperbolicity of G. An experimental evaluation of our methodology is provided for large collaboration networks. Finally, we deduce from our theoretical results the first linear-time algorithm to compute the hyperbolicity of an outerplanar graph. , pour tout quadruplet u, x, v, y de G. Nous étudions la relation entre l'hyperbolicité d'un graphe et celle de chacun de ses atomes. Ces derniers sont les sous-graphes résultant de la décompo-sition d'un graphe par des cliques-séparatrices [34,45]. Plus précisemment, nous montrons que l'hyperbolicité d'un atome est au plus l'hyperbolicité de G moins un. Nous proposons un algorithme pour modifier les atomes de sorte que la valeur maximale de l'hyperbolicité de ces atomes modifiés soit exactement l'hyperbolicité de G. La complexité de cet algorithme est la même que celle de la décomposition du graphe par des cliques-séparatrices. Nous évaluons expériementale-ment cette méthode sur des graphes de collaborations (co-auteurs). Enfin, nous proposons un algorithme pour calculer en temps linéaire l'hyperbolicité des graphes planaires extérieurs.

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Section: Definitions and Notationsmentioning

confidence: 99%

Applying clique-decomposition for computing Gromov hyperbolicity

Cohen

Coudert

Ducoffe

et al. 2017

Theoretical Computer Science

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“…These properties guarantee that G is a geodesic metric space and that G M can be defined. Note that excluding multiple edges and loops is not an important loss of generality, since ( [57], Theorems 8 and 10) they reduce the problem of computing the hyperbolicity constant of graphs with multiple edges and/or loops to the study of simple graphs.…”

Section: Definitions and Backgroundmentioning

confidence: 99%

Gromov Hyperbolicity in Mycielskian Graphs

et al. 2017

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Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph G M is hyperbolic and that δ(G M ) is comparable to diam(G M ). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that 5/4 ≤ δ(G M ) ≤ 5/2. Graphs G whose Mycielskian have hyperbolicity constant 5/4 or 5/2 are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on δ(G) just in terms of δ(G M ) is obtained.

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“…For each network, we identify a bi-connected component with the maximum value of since the hyperbolicity of a graph equals the maximum hyperbolicity of its bi-connected components. 8,22,53 Table 2 shows that almost all networks in our datasets have small hyperbolicity. Even though the definition of -hyperbolicity considers the difference between the largest two distance sums among any quadruples and takes into account only the maximum one, this absolute analysis is deficient.…”

Section: Hyperbolicity Of Biological Networkmentioning

confidence: 99%

Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

Alrasheed

Dragan

2016

Journal of Algorithms & Computational Technology

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Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov's notion of -hyperbolicity but also by analyzing its relationship to other graph's parameters. This new perspective allows us to classify graphs with respect to their hyperbolicity and to show that many biological networks are hyperbolic. Then we introduce the eccentricity-based bending property which we exploit to identify the core vertices of a graph by proposing two models: the maximum-peak model and the minimum cover set model. In this extended version of the paper, we include some new theorems, as well as proofs of the theorems proposed in the conference paper. Also, we present the algorithms we used for each of the proposed core identification models, and we provide more analysis, explanations, and examples.

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Gromov hyperbolic graphs

Cited by 61 publications

References 23 publications

Applying clique-decomposition for computing Gromov hyperbolicity

Applying clique-decomposition for computing Gromov hyperbolicity

Gromov Hyperbolicity in Mycielskian Graphs

Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

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