The metric dimension of the lexicographic product of graphs

Saputro, Suhadi Wido; Simanjuntak, Rinovia; Uttunggadewa, Saladin; Assiyatun, Hilda; Baskoro, Edy Tri; Salman, A. N. M.; Bača, Martin

doi:10.1016/j.disc.2013.01.021

Cited by 65 publications

(41 citation statements)

References 8 publications

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“…Also generalized Petersen graphs P(n, 3) have bounded metric dimension [12]. The metric dimension of the lexicographic product of graphs has been studied in [20]. The graphs having metric dimension 1 are characterized in the following theorem.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

“…A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension or location number of G, denoted by β (G) [5]. The concepts of resolving set and metric basis have previously appeared in the literature (see [1][2][3][4][5][6][7][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). …”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

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Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Naeem¹,

Imran²

2014

Appl. Math. Inf. Sci.

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Abstract:Metric dimension or location number is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs G n : F = (G n ) n≥1 depending on n as follows: the order |V (G)| = ϕ(n) and lim n→∞ ϕ(n) = ∞. If there exists a constant C > 0 such that dim(G n ) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), F is called a family with constant metric dimension.In this paper, we study the metric dimension of quasi flower snarks, generalized antiprism and cartesian product of square cycle and path. We prove that these classes of graphs have constant or bounded metric dimension. It is natural to ask for characterization of graphs classes with respect to the nature of their metric dimension. It is also shown that the exchange property of the bases in a vector space does not hold for minimal resolving sets of quasi flower snarks, generalized prism and generalized antiprism.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Naeem¹,

Imran²

2014

Appl. Math. Inf. Sci.

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“…Applications of the metric dimension to the navigation of robots in networks are discussed in [4] and applications to chemistry in [5,6]. This invariant was studied further in a number of other papers including, for instance [7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

The Simultaneous Local Metric Dimension of Graph Families

et al. 2017

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Abstract:In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if v, y). A set S ⊆ V is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set S ⊆ V is said to be a simultaneous local metric generator for a graph family G = {G 1 , G 2 , . . . , G k }, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of G. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter.

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“…The notion of lexicographic product was brought to the author by Professor Edy Tri Baskoro through his work [11] and that of strong product by Professor Akihiro Munemasa. The author thanks them for interesting conversation and for their instructing references.…”

Section: Acknowledgementsmentioning

confidence: 99%

Quantum Probability Aspects to Lexicographic and Strong Products of Graphs

Obata

2016

IIS

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The adjacency matrix of the lexicographic product of graphs is decomposed into a sum of monotone independent random variables in a certain product state. The adjacency matrix of the strong product of graphs admits an expression in terms of commutative independent random variables in a product state. Their spectral distributions are obtained by using the monotone, classical and Mellin convolutions of probability distributions.KEYWORDS: adjacency matrix, convolution of probability distributions, lexicographic product, spectral distribution, strong product Products of GraphsA graph G ¼ ðV; EÞ is a pair, where V is a non-empty set of vertices and E a set of edges, i.e., a subset of unordered pairs of distinct vertices. If fx; yg 2 E, we say that x and y are adjacent and write x $ y. We deal with both finite and infinite graphs, but always assume that a graph is locally finite, i.e., degðxÞ < 1 for all vertices x 2 V. The adjacency matrix of G, denoted by A ¼ A½G, is a matrix with index set V Â V defined byGiven two graphs G 1 ¼ ðV 1 ; E 1 Þ and G 2 ¼ ðV 2 ; E 2 Þ there is a large variety of forming their product to obtain a larger graph, see e.g., [4] and references cited therein. From the quantum probability viewpoint we have so far studied the Cartesian, star, comb and free products of graphs [1,6,9,10]. In this paper, being based on a similar spirit, we will discuss the lexicographic and strong products of graphs, and derive their spectral distributions using certain concepts of independence in quantum probability. Definition 1.1. The lexicographic product of G 1 and G 2 , denoted by G 1 B L G 2 , is the graph on V ¼ V 1 Â V 2 , where two distinct vertices ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ are adjacent whenever (i) x 1 $ x 2 ; or (ii) x 1 ¼ x 2 and y 1 $ y 2 . Lemma 1.2. Let G 1 and G 2 be graphs with adjacency matrices A 1 and A 2 , respectively. Then the adjacency matrix of the lexicographic productwhere J 2 is the matrix with index set V 2 Â V 2 whose entries are all one, and I 1 is the identity matrix with index setThe proof is straightforward by definition and is omitted. The Cartesian product of G 1 and G 2 , denoted by G 1 Â C G 2 , is the graph on V ¼ V 1 Â V 2 , where two distinct vertices ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ are adjacent whenever (i) x 1 ¼ x 2 and y 1 $ y 2 ; or (ii) x 1 $ x 2 and y 1 ¼ y 2 . The adjacency matrix of G 1 Â C G 2 is given byð1:2ÞThe Cartesian product is associative and commutative. By definition, G 1 Â C G 2 is a subgraph of G 1 B L G 2 , which is viewed also from the adjacency matrices ð1:1Þ and ð1:2Þ.Definition 1.3. The strong product of G 1 and G 2 , denoted by G 1 Â S G 2 , is the graph on V ¼ V 1 Â V 2 , where two distinct vertices ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ are adjacent whenever (i) x 1 ¼ x 2 or x 1 $ x 2 ; and (ii) y 1 ¼ y 2 or y 1 $ y 2 .

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The metric dimension of the lexicographic product of graphs

Cited by 65 publications

References 8 publications

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

The Simultaneous Local Metric Dimension of Graph Families

Quantum Probability Aspects to Lexicographic and Strong Products of Graphs

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