Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications

Tillquist, Richard C.; Frongillo, Rafael M.; Lladser, Manuel E.

doi:10.1137/21m1409512

Cited by 7 publications

(3 citation statements)

References 161 publications

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“…For a simple, connected, and undirected graph G = (V, E), the idea of metric dimension is just comparable to the number of satellites (in total) required for the perfect operation of GPS (Global Positioning Systems). The key concept behind metric dimension, is that a small ordered set, say U, is selected from the vertex set V(G), and the members of U are called as satellites or landmarks [18,19]. Then, using the distance of all vertices from the landmarks, one can uniquely identify each vertex in V(G) corresponding to the set U.…”

Section: Open Accessmentioning

confidence: 99%

“…For instance, the small set U can assist robots navigating in a physical space, identifying intersection points on roads, uniquely specifying a group of people by assigning a UIN (unique identification number), or tracing disease transmission between distinct regions [9,20]. This might be employed as well for more general activities like recognizing a source of lies and deceitful information in a network of people, comparing distinct architectures of networks, chemical structure categorization, or quantitatively encoding symbolic data [19].…”

Section: Open Accessmentioning

confidence: 99%

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Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Al-Qudah,

Jaradat,

Sharma

et al. 2024

Phys. Scr.

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For an undirected connected graph $G=G(V,E)$ with vertex set $V(G)$ and edge set $E(G)$, a subset $R$ of $V$ is said to be a resolving in $G$, if each pair of vertices (say $a$ and $b$; $a\neq b$) in $G$ satisfy the relation $d(a, k)\neq d(b, k)$, for at least one member $k$ in $R$. The minimum set $R$ with this resolving property is said to be a metric basis for $G$, and the cardinality of such set $R$, is referred to as the metric dimension of $G$, denoted by $dim_v(G)$. In this manuscript, we consider a complex molecular graph of one-heptagonal carbon nanocone (represented by $HCN_{s}$) and investigate its metric basis as well as metric dimension. We prove that just three specifically chosen vertices are enough to resolve the molecular graph of $HCN_{s}$. Moreover, several theoretical as well as applicative properties including comparison have also been incorporated.

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Section: Open Accessmentioning

confidence: 99%

Section: Open Accessmentioning

confidence: 99%

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Al-Qudah,

Jaradat,

Sharma

et al. 2024

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…Furthermore, this topic has some applications to problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures. Moreover, a survey has recently appeared [15] which contains the vast majority of the most important results on metric dimension of graphs.…”

Section: Introductionmentioning

confidence: 99%