Locating-Domination and Identification

Lobstein, Antoine; Hudry, Olivier; Charon, Irène

doi:10.1007/978-3-030-51117-3_8

Topics in Domination in Graphs

2020

DOI: 10.1007/978-3-030-51117-3_8

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Locating-Domination and Identification

Antoine Lobstein

Olivier Hudry

Irène Charon

Abstract: Locating-domination and identification are two particular, related, types of domination: a set C of vertices in a graph G = (V, E) is a locating-dominating code if it is dominating and any two vertices of V \ C are dominated by distinct sets of codewords; C is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This chapter presents a survey of the major results on locating-domination and on identification.

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“…The minimum cardinality of such set is called the k-locating-domination number denoted by LD k (G). Results about the k-locating-domination number can be found surveyed in [27]. Necessarily every k-locating-dominating set is a distance k-resolving dominating set, the opposite is not true.…”

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confidence: 99%

A study of a combination of distance domination and resolvability in graphs

Retnowardani¹,

Utoyo²,

Dafik³

et al. 2024

Discuss. Math. Graph Theory

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For k ≥ 1, in a graph G = (V, E), a set of vertices D is a distance kdominating set of G, if any vertex in V \ D is at distance at most k from some vertex in D. The minimum cardinality of a distance k-dominating set of G is the distance k-domination number, denoted by γ k (G). An ordered set of vertices W = {w 1 , w 2 , . . . , w r } is a resolving set of G, if for any two distinct vertices x and y in V \ W , there exists 1 ≤ i ≤ r such that 1052 D.A. Retnowardani, M.I.Utoyo, Dafik, L.Susilowati andK. DliouThe minimum cardinality of a resolving set of G is the metric dimension of the graph G, denoted by dim(G). In this paper, we introduce the distance k-resolving dominating set which is a subset of V that is both a distance k-dominating set and a resolving set of G. The minimum cardinality of a distance k-resolving dominating set of G is called the distance k-resolving domination number and is denoted by γ r k (G). We give several bounds for γ r k (G), some in terms of the metric dimension dim(G) and the distance k-domination number γ k (G). We determine γ r k (G) when G is a path or a cycle. Afterwards, we characterize the connected graphs of order n having γ r k (G) equal to 1, n − 2, and n − 1, for k ≥ 2. Then, we construct graphs realizing all the possible triples (dim(G), γ k (G), γ r k (G)), for all k ≥ 2. Later, we determine the maximum order of a graph G having distance k-resolving domination number γ r k (G) = γ r k ≥ 1, we provide graphs achieving this maximum order for any positive integers k and γ r k . Then, we establish Nordhaus-Gaddum bounds for γ r k (G), for k ≥ 2. Finally, we give relations between γ r k (G) and the k-truncated metric dimension of graphs and give some directions for future work.

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mentioning

confidence: 99%

A study of a combination of distance domination and resolvability in graphs

Retnowardani¹,

Utoyo²,

Dafik³

et al. 2024

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

show abstract

Codes in the q-ary Lee Hypercube

Charon

Hudry

Lobstein

2022

Wseas Transactions on Mathematics

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Locating-Domination and Identification

Cited by 2 publications

References 132 publications

A study of a combination of distance domination and resolvability in graphs

A study of a combination of distance domination and resolvability in graphs

Codes in the q-ary Lee Hypercube

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