The (Weighted) Metric Dimension of Graphs: Hard and Easy Cases

Epstein, Leah; Levin, Asaf; Woeginger, Gerhard J.

doi:10.1007/s00453-014-9896-2

Cited by 59 publications

(78 citation statements)

References 16 publications

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“…We refer to e.g. [1,12,16,17,18,19,21,29,31] for some results. On the positive side, Identifying Code, Locating-Dominating-Set and Open Locating-Dominating Set are linear-time solvable for graphs of bounded clique-width (using Courcelle's theorem [15] ).…”

Section: Similarly a Graph Admits An Open Locating-dominating Set Ifmentioning

confidence: 99%

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Foucaud

Mertzios

Naserasr

et al. 2017

Theoretical Computer Science

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We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a solution set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs. * u, v if it (totally) dominates exactly one of them. A set S (totally) separates the vertices of a set X if all pairs of X are (totally) separated by a vertex of S. Whenever it is clear from the context, we will only say "separate" and omit the word "totally". We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [33,34]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S. The smallest size of a locating-dominating set of G is the location-domination number of G, denoted γ LD (G). Without the domination constraint, this concept has also been used under the name distinguishing set in [2] and sieve in [28].Definition 2 (Karpovsky, Chakrabarty and Levitin [27]). A set S of vertices of a graph G is an identifying code if it is a dominating set and it separates all vertices of V (G). The smallest size of an identifying code of G is the identifying code number of G, denoted γ ID (G). Definition 3 (Seo and Slater [31]). A set S of vertices of a graph G is an open locating-dominating set if it is a total dominating set and it totally separates all vertices of V (G). The smallest size of an open locating-dominating set of G is the open location-domination number of G, denoted γ OLD (G). This concept has also been called identifying open code in [25]. Separation could also be done using distances from the members of the solution set. Let d(x, u) denote the distance between two vertices x and u. Definition 4 (Harary and Melter [24], Slater [32]). A set B of vertices of a graph G is a resolving set if for each pair u, v of distinct vertices, there is a vertex x of B with d(x, u) = d(x, v). 1 The smallest size of a resolving set of G is the metric dimension of G, denoted dim(G).

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Section: Similarly a Graph Admits An Open Locating-dominating Set Ifmentioning

confidence: 99%

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Foucaud

Mertzios

Naserasr

et al. 2017

Theoretical Computer Science

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“…In this work, Diaz et al also showed that this problem is solvable in polynomial time on the class of outer-planar graphs. Subsequently, Epstein et al [7] showed that this problem is NPcomplete on split graphs, bipartite and co-bipartite graphs. Subsequently, Epstein et al [7] showed that this problem is NPcomplete on split graphs, bipartite and co-bipartite graphs.…”

Section: Metric Dimensionmentioning

confidence: 99%

Metric Dimension of Bounded Width Graphs

Belmonte

Fomin

Golovach

et al. 2015

Mathematical Foundations of Computer Science 2015

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The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud et al. (2015) showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph.

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“…As further recent work related to this paper, we remark that the metric dimension of t-trees has recently been investigated in [5], and the treelength of a graph has recently been used to design algorithms to compute the metric dimension [6]. Algorithms and complexity results regarding the computation of the metric dimension of graphs belonging to graph classes considered in the present paper, can be found in [14,16,18].…”

Section: Introductionmentioning

confidence: 89%

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

Beaudou¹,

Dankelmann²,

Foucaud³

et al. 2018

SIAM J. Discrete Math.

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The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order n of a graph in terms of its diameter d and metric dimension k. In general, the bound n ≤ d k + k is known to hold. We prove a bound of the form n = O(kd 2 ) for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width w and length ℓ, we obtain a bound of the form n = O(kd 2 (2ℓ + 1) 3w+1 ). This implies in particular that n = O(kd O(1) ) for graphs of constant treewidth and n = O(f (k)d 2 ) for chordal graphs, where f is a doubly-exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomassé) as a tool, we prove the bounds n ≤ (dk + 1) t−1 + 1 for Kt-minor-free graphs, and n ≤ (dk + 1) d(3·2 r +2) + 1 for graphs of rankwidth at most r. *

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The (Weighted) Metric Dimension of Graphs: Hard and Easy Cases

Cited by 59 publications

References 16 publications

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Metric Dimension of Bounded Width Graphs

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

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