On the metric dimension of some families of graphs

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

doi:10.1016/j.endm.2005.06.023

Cited by 113 publications

(41 citation statements)

References 4 publications

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“…Under our propagation assumptions, this problem actually represents a known problem of finding a metric basis of a graph [6] or determining a resolving set of minimum cardinality [7]. For an arbitrary graph, this problem is NP-hard [6], but explicit results exist for some families of graphs [8]. Here we review some of the results in the context of selecting the smallest subset of nodes for observation and illustrate the performance of the available approximation algorithm.…”

Section: Introductionmentioning

confidence: 97%

Network observability and localization of the source of diffusion based on a subset of nodes

Zejnilovic

Gomes

Sinopoli

2013

2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Identifying the patient-zero of an epidemic outbreak, locating the person who started a rumor in a social network, finding the computer that initiated the spreading of a computer virus in a network-these are all applications of localizing the source of diffusion in a network. Since most of the networks of interest are very large, we are usually able to observe only a part of the network. In this paper, we first present a model for the dynamics of network diffusion similar to state update of a linear time-varying system. Based on this model, we provide a sufficient condition for observability of the network, i.e., we establish when is the partial information available to us sufficient to uniquely localize the source. Also, we connect the problem of finding the smallest subset of observed nodes to the problem of metric basis of the graph. We then present different methods to perform source localization depending on network observability.

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Section: Introductionmentioning

confidence: 97%

Network observability and localization of the source of diffusion based on a subset of nodes

Zejnilovic

Gomes

Sinopoli

2013

2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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“…Interested reader is referred e.g. to [23]. In [26] it was proved that the problem of computing the metric dimension of an arbitrary graph is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

Computing the metric dimension of graphs by genetic algorithms

2007

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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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On the metric dimension of some families of graphs

Cited by 113 publications

References 4 publications

Network observability and localization of the source of diffusion based on a subset of nodes

Network observability and localization of the source of diffusion based on a subset of nodes

Computing the metric dimension of graphs by genetic algorithms

The Simultaneous Local Metric Dimension of Graph Families

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