The distance-k dimension of graphs

Geneson, Jesse; Yi, Eunjeong

doi:10.48550/arxiv.2106.08303

Cited by 4 publications

(8 citation statements)

References 14 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let S be any minimum distance-k resolving set of G. First, suppose that |S| is odd. As shown in [9], there exists a minimum distance-k resolving set S 0 of G such that S 0 is also a distance-k dominating set of G. So, γ k L (G) = dim k (G). Second, suppose |S| is even.…”

Section: γ K L (G) Of Some Classes Of Graphsmentioning

confidence: 98%

“…For a construction of graphs G with dim 2 (G) = β of maximum order β + 3 β , we refer to [9]. For k ∈ {1, 2}, the same construction of graphs G by removing the vertex w with code S,k (w) = (k+1) |S| , where S is a minimum distance-k resolving set of G, provides graphs with the maximum order of G in Corollary 3.9.…”

Section: And Equality Holds If and Only Ifmentioning

confidence: 99%

“…Second, suppose |S| is even. If n ≡ 1 (mod 3k + 2), then there exists a minimum distance-k resolving set S 1 of G such that S 1 is also a distance-k dominating set of G (see [9]); thus, γ k L (G) = dim k (G). If n ≡ 1 (mod 3k + 2), then there exists a vertex w in G with code S,k (w) = (k+1) |S| for any minimum distance-k resolving set S of G (see [9]); thus, γ k L (G) = dim k (G) + 1.…”

Section: γ K L (G) Of Some Classes Of Graphsmentioning

confidence: 99%

“…We recall how the distance-k dimension of a graph changes upon deletion of an edge. (c) [9] for k ≥ 3, dim k (G − e) ≤ dim k (G) + 2;…”

Section: γ K L (G) Of Some Classes Of Graphsmentioning

confidence: 99%

“…z a e Figure 4: [9] Graphs G such that dim k (G) − dim k (G − e) can be arbitrarily large, where k ≥ 2 and a ≥ 3.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Distance-$k$ locating-dominating sets in graphs

Kang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let G be a graph with vertex set V , and let k be a positive integer. A set D ⊆ V is a distance-k dominating set of G if, for each vertex u ∈ V − D, there exists a vertex w ∈ D such that d(u, w) ≤ k, where d(u, w) is the length of a shortest path between the vertices in, of G is the minimum cardinality of all sets S ⊆ V such that S is both a distance-k dominating set and a distance-k resolving set of G. Note that γ 1 L (G) is the well-known location-domination number introduced by Slater in 1988. For any connected graph G of order n ≥ 2, we obtain the following sharp bounds:γ k (G) can be arbitrarily large. Moreover, for any tree T of order n ≥ 2, we show that γ k L (T ) ≤ n − ex(T ), where ex(T ) denotes the number of exterior major vertices of T , and we characterize trees T achieving equality. We also examine the effect of edge deletion on the distance-k location-domination number of graphs.

show abstract

Section: γ K L (G) Of Some Classes Of Graphsmentioning

confidence: 98%

Section: And Equality Holds If and Only Ifmentioning

confidence: 99%

Section: γ K L (G) Of Some Classes Of Graphsmentioning

confidence: 99%

“…We recall how the distance-k dimension of a graph changes upon deletion of an edge. (c) [9] for k ≥ 3, dim k (G − e) ≤ dim k (G) + 2;…”

Section: γ K L (G) Of Some Classes Of Graphsmentioning

confidence: 99%

“…z a e Figure 4: [9] Graphs G such that dim k (G) − dim k (G − e) can be arbitrarily large, where k ≥ 2 and a ≥ 3.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Distance-$k$ locating-dominating sets in graphs

Kang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Truncated Metric Dimension for Finite Graphs

Tillquist¹,

Frongillo²,

Lladser³

2021

Preprint

View full text Add to dashboard Cite

A graph G = (V, E) with geodesic distance d(•, •) is said to be resolved by a non-empty subset R of its vertices when, for all vertices u and v, if d(u, r) = d(v, r) for each r ∈ R, then u = v. The metric dimension of G is the cardinality of its smallest resolving set. In this manuscript, we present and investigate the notions of resolvability and metric dimension when the geodesic distance is truncated with a certain threshold k; namely, we measure distances in G using the metric d k (u, v) := min{d(u, v), k +1}. We denote the metric dimension of G with respect to d k as β k (G). We study the behavior of this quantity with respect to k as well as the diameter of G. We also characterize the truncated metric dimension of paths and cycles as well as graphs with extreme metric dimension, including graphs of order n such that β k (G) = n − 2 and β k (G) = n − 1. We conclude with a study of various problems related to the truncated metric dimension of trees.

show abstract

Sharp bound on the threshold metric dimension of trees

Bartha¹,

Komjáthy²,

Raes³

2021

Preprint

View full text Add to dashboard Cite

The threshold-k metric dimension (Tmd k ) of a graph is the minimum number of sensors -a subset of the vertex set -needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the measuring radius of a sensor is k. We give a sharp lower bound on the Tmd k of trees, depending only on the number of vertices n and the measuring radius k. This sharp lower bound grows linearly in n with leading coefficient 3/(k 2 + 4k + 3 + 1{k ≡ 1 (mod 3)}), disproving earlier conjectures by Tillquist et al. in [43] that suspected n/( k 2 /4 +2k) as main order term. We provide a construction for the largest possible trees with a given Tmd k value. The proof that our optimal construction cannot be improved relies on edge-rewiring procedures of arbitrary (suboptimal) trees with arbitrary resolving sets, which reveal the structure of how small subsets of sensors measure and resolve certain areas in the tree that we call the attraction of those sensors. The notion of 'attraction of sensors' might be useful in other contexts beyond trees to solve related problems. We also provide an improved lower bound on the Tmd k of arbitrary trees that takes into account the structural properties of the tree, in particular, the number and length of simple paths of degree-two vertices terminating in leaf vertices. This bound complements [43], where only trees without degree-two vertices were considered, except the simple case of a single path.

show abstract

The distance-k dimension of graphs

Cited by 4 publications

References 14 publications

Distance-$k$ locating-dominating sets in graphs

Distance-$k$ locating-dominating sets in graphs

Truncated Metric Dimension for Finite Graphs

Sharp bound on the threshold metric dimension of trees

Contact Info

Product

Resources

About