Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

Gravier, Sylvain; Parreau, Aline; Rottey, Sara; Storme, Leo; Vandomme, Elise

doi:10.37236/5256

Cited by 10 publications

(7 citation statements)

References 33 publications

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“…For more information about distance-regular graphs, see the book of Brouwer, Cohen and Neumaier [12] and the forthcoming survey paper by van Dam, Koolen and Tanaka [16]. In recent years, a number of papers have been written on the subject of the metric dimension of distance-regular graphs (and on the related problem of class dimension of association schemes), by the present author and others: see [5,6,7,8,10,14,17,22,23,24,25,26,29,34], for instance. In this paper, we shall focus on various classes of imprimitive distance-regular graphs, which are explained below.…”

Section: Introductionmentioning

confidence: 97%

On the Metric Dimension of Imprimitive Distance-Regular Graphs

Bailey

2016

Ann. Comb.

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A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distanceregular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs, but also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.

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Section: Introductionmentioning

confidence: 97%

On the Metric Dimension of Imprimitive Distance-Regular Graphs

Bailey

2016

Ann. Comb.

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“…Honkala and Laihonen [14] studied identifying codes in the king grid that are robust against edge deletions. More recently, identifying codes have been considered for vertex-transitive graphs and strongly regular graphs by Gravier et al [13], and for graphs of girth at least five by Balbuena, Foucaud and Hansberg [2].…”

Section: Introductionmentioning

confidence: 99%

Sufficient conditions for a digraph to admit a (1,≤ l)-identifying code

Balbuena¹,

Dalfó²,

Martínez-Barona³

2021

Discuss. Math. Graph Theory

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A (1, ≤ )-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most have distinct closed inneighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree δ − ≥ 1 to admit a (1, ≤ )-identifying code for = δ − , δ − + 1. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree δ ≥ 2 and girth at least 7 admits a (1, ≤ δ)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, ≤ 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, ≤ )-identifying code for = 2, 3.Mathematics Subject Classifications: 05C69, 05C20

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“…For this purpose Karpovsky, Chakrabarty and Levitin defined identifying codes in [17]. More on identifying codes can be found at [20] and for recent development, see [1], [8] and [12]. for each pair of distinct vertices u, v ∈ V .…”

Section: Introductionmentioning

confidence: 99%

On a Conjecture Regarding Identification in Hamming Graphs

Junnila

Laihonen

Lehtilä

2019

Electron. J. Combin.

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Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In particular, there are a lot of results regarding the binary hypercubes, that is, the Hamming graphs K n 2 . In 2008, Gravier et al. started investigating identification in K 2 q . Goddard and Wash, in 2013, studied identifying codes in the general Hamming graphs K n q . They stated, for instance, that γ ID (K n q ) ≤ q n−1 for any q and n ≥ 3. Moreover, they conjectured that γ ID (K 3 q ) = q 2 . In this article, we show that γ ID (K 3 q ) ≤ q 2 − q/4 when q is a power of four, disproving the conjecture. Our approach is based on the recursive use of suitable designs. Goddard and Wash also gave the following lower boundThe conventional methods used for obtaining lower bounds on identifying codes do not help here. Hence, we provide a different technique building on the approach of Goddard and Wash. Moreover, we improve the above mentioned bound γ ID (K n q ) ≤ q n−1 to γ ID (K n q ) ≤ q n−k for n = 3 q k −1 q−1 when q is a prime power. For this bound, we utilize suitable linear codes over finite fields and a class of closely related codes, namely, the self-locating-dominating codes. In addition, we show that the self-locating-dominating codes satisfy the result γ SLD (K 3 q ) = q 2 related to the above conjecture. Figure 1: Optimal identifying, self-identifying and self-locating-dominating codes in G. Definition 3. A code C ⊆ V is called self-identifying in G if the code C is identifying in G and for all u ∈ V and U ⊆ V such that |U | ≥ 2 we have I(C; u) = I(C; U ).A self-identifying code C in a finite graph G with the smallest cardinality is called optimal and the number of codewords in an optimal self-identifying code is denoted by γ SID (G).

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Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

Cited by 10 publications

References 33 publications

On the Metric Dimension of Imprimitive Distance-Regular Graphs

On the Metric Dimension of Imprimitive Distance-Regular Graphs

Sufficient conditions for a digraph to admit a (1,≤ l)-identifying code

On a Conjecture Regarding Identification in Hamming Graphs

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