Gromov Hyperbolicity in Mycielskian Graphs

Granados, Ana; Pestana, Domingo; Portilla, Ana; Rodrı́guez, José M.

doi:10.3390/sym9080131

Cited by 7 publications

(4 citation statements)

References 54 publications

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“…Mycielskian graphs are always hyperbolic [71]. Now, we summarize the known results about the hyperbolicity of the main class of binary operations in graphs: products of graphs.…”

Section: Introductionmentioning

confidence: 97%

“…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%

See 1 more Smart Citation

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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“…Mycielskian graphs are always hyperbolic [71]. Now, we summarize the known results about the hyperbolicity of the main class of binary operations in graphs: products of graphs.…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Symmetry in Graph Theory

2019

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“…Moreover, they determined the generalized 3-connectivity of the Mycielskian graph of the tree T n , the complete graph K n and the complete bipartite graph K a,b . Now, the results on some graph parameters of the Mycielskian graphs have been obtained, see [29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Generalized Connectivity of the Mycielskian Graph under g-Extra Restriction

Zou,

Li,

Zhang

et al. 2023

Mathematics

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The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset S⊆V is called a g-extra cut of G if the graph induced by the set G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G, denoted as κg(G), is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ(G), also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph μ(G) and the graph G. In addition, we show that κ3(μ(G))=2κ1(G)+1 for κ1(G)≤min{4,⌊n2⌋}, and prove the bounds of κ2g+1(μ(G)) for g≥2.

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“…This construction was introduced by Mycielski in [13] in order to construct triangle-free graphs and arbitrarily large chromatic number. As a construction of independent interest, several other graph parameters of the Mycielskian construction have been investigated, for example dominator colouring number [1], connectivity [3], energy [4], L(2, 1)-labelling number [7], circular chromatic number [9], Gromov hyperbolicity [11], wide diameter [16] and Italian domination number [19], amongst many others.…”

Section: Introductionmentioning

confidence: 99%

On the General Position Number of Mycielskian Graphs

Thomas¹,

Chandran²,

Tuite³

et al. 2022

Preprint

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The general position problem for graphs was inspired by the no-three-in-line problem and the general position subset selection problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path between two vertices of S contains a third element of S; the general position number of G is the size of a largest general position set. In this note we investigate the general position numbers of the Mycielskian of graphs, which are of interest in chromatic graph theory. We give tight upper and lower bounds on the general position number of the Mycielskian of a graph G and investigate the structure of the graphs meeting these bounds. We determine this number exactly for common classes of graphs including cubic graphs and a wide range of trees.

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Gromov Hyperbolicity in Mycielskian Graphs

Cited by 7 publications

References 54 publications

Symmetry in Graph Theory

Symmetry in Graph Theory

Generalized Connectivity of the Mycielskian Graph under g-Extra Restriction

On the General Position Number of Mycielskian Graphs

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