Fixing Numbers of Graphs and Groups

Gibbons, Courtney R.; Laison, Joshua D.

doi:10.37236/128

Cited by 22 publications

(15 citation statements)

References 13 publications

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“…In [58], Gibbons and Laison consider some problems about bases of automorphism groups of graphs (although they used the terms fixing set and fixing number). They give a greedy algorithm for finding bases (build a base by successively choosing vertices from the largest orbit of the stabilizer of the points previously chosen).…”

Section: The Symmetric Group Johnson and Kneser Graphs And The Greementioning

confidence: 99%

Base size, metric dimension and other invariants of groups and graphs

Bailey¹,

Cameron²

2011

Bulletin of the London Mathematical Society

223

297

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The base size of a permutation group, and the metric dimension of a graph, are two of a number of related parameters of groups, graphs, coherent configurations and association schemes. They have been repeatedly redefined with different terminology in various different areas, including computational group theory and the graph isomorphism problem. We survey results on these parameters in their many incarnations, and propose a consistent terminology for them. We also present some new results, including on the base sizes of wreath products in the product action, and on the metric dimension of Johnson and Kneser graphs.

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Section: The Symmetric Group Johnson and Kneser Graphs And The Greementioning

confidence: 99%

Base size, metric dimension and other invariants of groups and graphs

Bailey¹,

Cameron²

2011

Bulletin of the London Mathematical Society

223

297

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“…For recent papers on determining sets see the works by Albertson, Boutin, Collins, Erwin, Gibbons, Harary, and Laison [1,4,5,9,11,12,13]. Determining sets are frequently used to identify the automorphism group of a graph.…”

Section: Introductionmentioning

confidence: 99%

On the Determining Number and the Metric Dimension of Graphs

Cáceres

Garijo

Puertas

et al. 2010

Electron. J. Combin.

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“…The minimum cardinality of a determining set is called the determining number. In [9], it was shown that fixing set and determining set are equivalent. A considerable literature has been developed in this field (see [4,9,11,15]).…”

Section: Introductionmentioning

confidence: 99%

“…In [9], it was shown that fixing set and determining set are equivalent. A considerable literature has been developed in this field (see [4,9,11,15]). The concept of fixing number originates from the idea of breaking symmetries in graphs, which have applications in the problem of programming a robot to manipulate objects [13].…”

Section: Introductionmentioning

confidence: 99%

On The Fixatic Number of Graphs

Fazil,

Javaid

2017

Preprint

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The fixing number of a graph G is the smallest cardinality of a set of vertices F ⊆ V (G) such that only the trivial automorphism of G fixes every vertex inThe cardinality of a largest fixatic partition is called the fixatic number of G. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper. N(u) − {v} = N(v) − {u}, then u and v are called twin vertices (or simply twins) in G. If for a vertex u of G, there exists a vertex v = u in G such that u, v are twins in G, then u is said to be a twin in G. A set T ⊆ V (G) is said to be a twin-set in G if any two of its elements are twins.An automorphism α of G, α : V (G) → V (G), is a bijective mapping such that α(u)α(v) ∈ E(G) if and only if uv ∈ E(G). Thus, each automorphism α of G is a permutation of the vertex set V (G) which preserves adjacencies and non-adjacencies. The automorphism group of a graph G, denoted by Γ(G), is the set of all automorphisms of a graphFor a vertex v of a graph G, the orbit of v, denoted O(v), is the set of all vertices {u ∈ V (G) : α(v) = u for some α ∈ Γ(G)}. Two vertices u and v are similar if they belong to the same orbit.A vertex v is fixed by a group element gis trivial. The fixing number of a graph G, denoted by f ix(G), is the smallest cardinality of a fixing set of G [8]. The graphs

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Fixing Numbers of Graphs and Groups

Cited by 22 publications

References 13 publications

Base size, metric dimension and other invariants of groups and graphs

Base size, metric dimension and other invariants of groups and graphs

On the Determining Number and the Metric Dimension of Graphs

On The Fixatic Number of Graphs

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