Distinguishing critical graphs

Alikhani, Saeid; Soltani, Samaneh

doi:10.1080/02522667.2020.1773614

Cited by 1 publication

(4 citation statements)

References 11 publications

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“…In [3], Alikhani and Soltani introduced the notion of a distinguishing critical graph. We reproduce their definition here:…”

Section: Distinguishing Critical Graphsmentioning

confidence: 99%

“…Example 6.5. We can obtain the automorphism group of D 4 through partitioning Sym 2 V in the following way: label (1, 2), (2, 3), (3,4), (4, 1) with the label 1 and all other basis elements with 2. We can also obtain it through partitioning X 2 in the following way: label (1, 2), (2, 1), (2, 3), (3, 2), (3,4), (4, 3), (4, 1), (1,4) with the label 1 and all other elements with 2.…”

Section: Structure Of Automorphism Groups Of Graphsmentioning

confidence: 99%

“…Example 6.6. We can obtain the cyclic group generated by (1234) by partitioning X 2 in in the following way: label (1, 2)(2, 3), (3,4), (4, 1) with the label 1 and all other elements with 2. This cannot be obtained as the automorphism group of an undirected graph.…”

Section: Structure Of Automorphism Groups Of Graphsmentioning

confidence: 99%

“…1 : (1, 2, 3), (1,3,4), (1,4,2), (2, 1, 4), (2, 3, 1), (2,4,3), (3, 1, 2), (3,2,4), (4, 1, 3), (4, 2, 1), (4, 3, 2) and 2 for everything else.…”

Section: Automorphism Groups Of Hypergraphsmentioning

confidence: 99%

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Distinguishing Numbers and Generalizations

2018

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The distinguishing number of a graph was introduced by Albertson and Collins in [1] as a measure of the amount of symmetry contained in the graph. Tymoczko extended this definition to faithful group actions on sets in [11]; taking the set to be the vertex set of a graph and the group to be the automorphism group of the graph allows one to recover the previous definition. Since then, several authors have studied properties of the distinguishing number as well as extensions of the notion. In this paper, we first answer a few open questions regarding the distinguishing number. Next we turn to generalizations regarding the labeling of Cartesian powers of a set and the different subgroups that can be obtained through labelings. We then introduce a new partially ordered set on partitions that follows naturally from extending the theory of distinguishing numbers to that of distinguishing partitions. Then we investigate the groups obtainable from partitioning Cartesian powers of a set in more detail and show how the original notion of the distinguishing number of a graph can be recovered in this way. Next, we introduce a polynomial and a symmetric function generalization of the distinguishing number. Finally, we present a large number of open questions and problems for further research.

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“…In [3], Alikhani and Soltani introduced the notion of a distinguishing critical graph. We reproduce their definition here:…”

Section: Distinguishing Critical Graphsmentioning

confidence: 99%

Section: Structure Of Automorphism Groups Of Graphsmentioning

confidence: 99%

Section: Structure Of Automorphism Groups Of Graphsmentioning

confidence: 99%

“…1 : (1, 2, 3), (1,3,4), (1,4,2), (2, 1, 4), (2, 3, 1), (2,4,3), (3, 1, 2), (3,2,4), (4, 1, 3), (4, 2, 1), (4, 3, 2) and 2 for everything else.…”

Section: Automorphism Groups Of Hypergraphsmentioning

confidence: 99%

See 2 more Smart Citations

Distinguishing Numbers and Generalizations

2018

Preprint

View full text Add to dashboard Cite

show abstract

Distinguishing critical graphs

Cited by 1 publication

References 11 publications

Distinguishing Numbers and Generalizations

Distinguishing Numbers and Generalizations

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