The distance 2-resolving domination number of graphs

Wardani, D. A. R.; Utoyo, Mohammad Imam; Dafik, Dafik; Dliou, Kamal

doi:10.1088/1742-6596/1836/1/012017

Cited by 1 publication

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“…This means that every superset of a distance k-resolving dominating set is also a distance k-resolving dominating set. We give the following bounds that extend bounds given for k equal to 1 and 2, in [5] and [34] respectively to all k ≥ 1.…”

Section: Preliminary Results and Bounds For γ R K (G)mentioning

confidence: 73%

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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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Section: Preliminary Results and Bounds For γ R K (G)mentioning

confidence: 73%

“…2k+1 [12]. The values of the distance k-resolving domination number of P n for k equal to 1 and 2 are given respectively in [4] and [34]. In the following we give γ r k (P n ) for all k ≥ 1.…”

Section: Preliminary Results and Bounds For γ R K (G)mentioning

confidence: 99%