Locating–dominating codes: Bounds and extremal cardinalities

Cáceres, José; Hernando, Carmen; Mora, Mercè; Pelayo, Ignacio M.; Puertas, María Luz

doi:10.1016/j.amc.2013.05.060

Cited by 24 publications

(39 citation statements)

References 15 publications

(12 reference statements)

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“…As a straight consequence of Proposition 4 (5), the following result is derived. [13] and [3] in S \ {1, 2} is {3}; and the neighborhood of vertices [1245] and [245] in S \ {1, 2} is {4, 5} (see Figure 3).…”

Section: The Graph Associated With a Distinguishing Setmentioning

confidence: 99%

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Locating domination in bipartite graphs and their complements

Hernando

Mora

Pelayo

2019

Discrete Applied Mathematics

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A set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G, λ(G), is the minimum cardinality of a locating-dominating set. In this work we study relationships between λ(G) and λ(G) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying λ(G) = λ(G) + 1. To this aim, we define an edge-labeled graph G S associated with a distinguishing set S that turns out to be very helpful.

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Section: The Graph Associated With a Distinguishing Setmentioning

confidence: 99%

“…[15] [13] [34] [3] [1] Proposition 5. Let S be a distinguishing set of cardinality k of a connected graph G of order n. Let G S be its associated graph.…”

Section: The Graph Associated With a Distinguishing Setmentioning

confidence: 99%

Locating domination in bipartite graphs and their complements

Hernando

Mora

Pelayo

2019

Discrete Applied Mathematics

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“…For the situations when the sensor can distinguish whether the anomaly is in the open neighbourhood of the sensor or in the location of the sensor itself, we have locating-dominating codes which were introduced by Slater in [21,24,25] (for recent developments, see [2] and [20]). More precisely, a code C ⊆ V is locating-dominating in G if the identifying sets I(C; u) are nonempty and unique for all u ∈ V \ C. Inspired by self-identifying codes, we may analogously define so called self-locating-dominating codes, which have been introduced and motivated in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Lemma 20. The code C L = {(2, 1, 3), (3, 1, 4), (4, 1, 2), (1, 2, 4), (3, 2, 1), (4, 2, 3), (1, 3, 2), (2, 3, 4), (4, 3, 1), (1,4,3), (2, 4, 1), (3,4,2)…”

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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“…Given a graph G, a set S ⊆ V (G) is a dominating set if every vertex not in S is adjacent to some vertex in S. A set S ⊆ V (G) is a locating-dominating set if S is a dominating set and N (u) ∩ S = N (v) ∩ S for every two different vertices u and v not in S. The location-domination number of G, denoted by λ(G), is the minimum cardinality of a locating-dominating set. In [8,27], bounds for this parameter are given. In this paper, merging the concepts studied in [11,39], we introduce the neighbor-locating colorings and the neighbor-locating chromatic number, and examine this parameter in some families of graphs.…”

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confidence: 99%

Neighbor-locating colorings in graphs

Alcón

Gutiérrez

Hernando

et al. 2020

Theoretical Computer Science

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A k-coloring of a graph G is a k-partition Π = {S 1 , . . . , S k } of V (G) into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u, v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number χ N L (G) is the minimum cardinality of a neighbor-locating coloring of G.We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n ≥ 5 with neighbor-locating chromatic number n or n − 1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.Henning and O. R. Oellermann introduced the so-called metric-locating-dominating sets, by merging the concepts of metric-locating set and dominating set.In [14], G. Chartrand, E. Salehi and P. Zhang, brought the notion of metric location to the ambit of vertex partitions, introducing the resolving partitions, also called metriclocating partition, and defining the partition dimension. Metric location and domination, in the context of vertex partitions, are studied in [28]. In [11], there were introduced the so-called locating colorings considering resolving partitions formed by independents sets.Neighbor location in sets was introduced by P. Slater in [39]. Given a graph G, a set S ⊆ V (G) is a dominating set if every vertex not in S is adjacent to some vertex in S. A set S ⊆ V (G) is a locating-dominating set if S is a dominating set and N (u) ∩ S = N (v) ∩ S for every two different vertices u and v not in S. The location-domination number of G, denoted by λ(G), is the minimum cardinality of a locating-dominating set. In [8,27], bounds for this parameter are given. In this paper, merging the concepts studied in [11,39], we introduce the neighbor-locating colorings and the neighbor-locating chromatic number, and examine this parameter in some families of graphs.

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Locating–dominating codes: Bounds and extremal cardinalities

Cited by 24 publications

References 15 publications

Locating domination in bipartite graphs and their complements

Locating domination in bipartite graphs and their complements

On a Conjecture Regarding Identification in Hamming Graphs

Neighbor-locating colorings in graphs

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