Minimal doubly resolving sets and the strong metric dimension of some convex polytopes

We consider the problem of determining the cardinality $\psi(H_{2,k})$ of minimal doubly resolving sets of Hamming graphs $H_{2,k}.$ We prove that for $k \geq 6$ every minimal resolving set of $H_{2,k}$ is also a doubly resolving set, and, consequently, $\psi(H_{2,k})$ is equal to the metric dimension of $H_{2,k},$ which is known from the literature. Moreover, we find an explicit expression for the strong metric dimension of all Hamming graphs $H_{n,k}.

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“…( [17]) If S is a strong resolving set of graph G, then, for every two vertices u, v ∈ V G such that d(u, v) = Diam(G), u ∈ S or v ∈ S.…”

Section: Strong Metric Dimension Of H Nkmentioning

confidence: 99%

“…In order to prove Theorem 4 we use the following lemma from [17], which is related to a strong resolving set of an arbitrary graph G.…”

Section: Strong Metric Dimension Of H Nkmentioning

confidence: 99%

Minimal doubly resolving sets and the strong metric dimension of Hamming graphs

Kratica

Kovačevíc-Vujčić²,

Čangalović³

et al. 2012

APPL ANAL DISCRETE M

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“…Property 1 (Kratica et al, 2012b) If S ⊂ V is a strong resolving set of graph G, then, for every two maximally distant vertices u, v ∈ V , it must be u ∈ S or v ∈ S.…”

Section: Introductionmentioning

confidence: 99%

The strong metric dimension of some generalized Petersen graphs

Kratica

Kovačevíc-Vujčić²,

Čangalović³

2017

APPL ANAL DISCRETE M

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“…The problem of finding the strong metric dimension of a graph has been studied for several classes of graphs. For instance, this problem was studied for Cayley graphs [21], distance-hereditary graphs [19], Hamming graphs [15], Cartesian product graphs and direct product graphs [24], corona product graphs and join graphs [16], strong product graphs [17,18] and convex polytopes [12] . Also, some Nordhaus-Gaddum type results for the strong metric dimension of a graph and its complement are known [28].…”

Section: Introductionmentioning

confidence: 99%

The Simultaneous Strong Metric Dimension of Graph Families

Estrada‐Moreno

García-Gómez

Ramírez-Cruz

et al. 2015

Bull. Malays. Math. Sci. Soc.

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Let ${\cal G}$ be a family of graphs defined on a common (labeled) vertex set $V$. A set $S\subset V$ is said to be a simultaneous strong metric generator for ${\cal G}$ if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for ${\cal G}$, denoted by $Sd_s({\cal G})$, is called the simultaneous strong metric dimension of ${\cal G}$. We obtain general results on $Sd_s({\cal G})$ for arbitrary families of graphs, with special emphasis on the case of families composed by a graph and its complement. In particular, it is shown that the problem of finding the simultaneous strong metric dimension of families of graphs is $NP$-hard, even when restricted to families of trees.Comment: arXiv admin note: text overlap with arXiv:1312.1987 by other author

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Minimal doubly resolving sets and the strong metric dimension of some convex polytopes

Cited by 60 publications

References 15 publications

Minimal doubly resolving sets and the strong metric dimension of Hamming graphs

Minimal doubly resolving sets and the strong metric dimension of Hamming graphs

The strong metric dimension of some generalized Petersen graphs

The Simultaneous Strong Metric Dimension of Graph Families

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