On the Hyperbolicity Constant in Graph Minors

Carballosa, Walter; Rodrı́guez, José M.; Rosario, Omar; Sigarreta, José M.

doi:10.1007/s41980-018-0032-y

Cited by 2 publications

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References 41 publications

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“…Therefore, it is useful to study hyperbolicity for particular classes of graphs. There are many works dealing with the hyperbolicity of different types of graphs: median graphs [53], line graphs [54][55][56], cubic graphs [57], complement graphs [58], regular graphs [59], chordal graphs [25,42,45,60], planar graphs [61,62], bipartite and intersection graphs [63], vertex-symmetric graphs [64], periodic graphs [65,66], expanders [34], bridged graphs [67], short graphs [68], graph minors [69], graphs with small hyperbolicity constant [70], Mycielskian graphs [71], geometric graphs [56,72], and some types of products of graphs: Cartesian product and sum [46,73], strong product [74], lexicographic product [75], and corona and join product [76].…”

Section: Introductionmentioning

confidence: 99%

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Symmetry in Graph Theory

2019

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This book contains the successful invited submissions [1-10] to a special issue of Symmetry on the subject area of 'graph theory'.Although symmetry has always played an important role in graph theory, in recent years, this role has increased significantly in several branches of this field, including, but not limited to: Gromov hyperbolic graphs, metric dimension of graphs, domination theory, and topological indices. This Special issue invites contributions addressing new results on these topics, both from a theoretical and an applied point of view.This special issue includes the novel techniques and tools for graph theory, such as:Published submissions are related to local metric dimension, Gromov hyperbolicity, differential, path ordinal method, neural networks, connectivity indices, and independent sets, as well as their applications.We found the edition and selections of papers for this book very inspiring and rewarding. We also thank the editorial staff and reviewers for their efforts and help during the process.

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Section: Introductionmentioning

confidence: 99%

“…For a large class of minor graphs, the minor graph is hyperbolic if and only if the original graph does [69].…”

Section: Introductionmentioning

confidence: 99%

Symmetry in Graph Theory

2019

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Hyperbolicity of Direct Products of Graphs

et al. 2018

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If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3 } is the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf{δ ≥ 0 : X is δ-hyperbolic }. Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product G 1 × G 2 is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain formulae or good bounds for the hyperbolicity constant of the direct product of some important graphs.

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On the Hyperbolicity Constant in Graph Minors

Cited by 2 publications

References 41 publications

Symmetry in Graph Theory

Symmetry in Graph Theory

Hyperbolicity of Direct Products of Graphs

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