Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

Foucaud, Florent; Mertzios, George B.; Naserasr, Reza; Parreau, Aline; Valicov, Petru

doi:10.1007/978-3-662-53174-7_32

Cited by 13 publications

(15 citation statements)

References 29 publications

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“…Contrary to what we claimed in the conference version of this paper [28], our reduction gadgets are not interval graphs and permutation graphs at the same time. Hence, we leave it as an open question to determine the complexity of the studied problems when restricted to graphs which are both interval and permutation graphs.…”

Section: Resultscontrasting

confidence: 81%

“…We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable. * A short version of this paper, containing only the results about location-domination and metric dimension, appeared in the proceedings of the WG 2015 conference [28].pair in X has a vertex in S (totally) separating it. We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [52,53], Babai [3]).…”

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confidence: 99%

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Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

et al. 2016

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We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W [2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable. * A short version of this paper, containing only the results about location-domination and metric dimension, appeared in the proceedings of the WG 2015 conference [28].pair in X has a vertex in S (totally) separating it. We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [52,53], Babai [3]). 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.

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Section: Resultscontrasting

confidence: 81%

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confidence: 99%

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

et al. 2016

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“…Deciding whether a given graph G has metric dimension at most k for a given integer k is known to be NP-complete for general graphs [11], planar graphs [5], even for those with maximum degree 6 and Gabriel unit disk graphs [17]. Epstein et al showed the NP-completeness for split graphs, bipartite graphs, co-bipartite graphs and line graphs of bipartite graphs [6] and Foucaud et al for permutation and interval graphs [9] [10].…”

Section: Introductionmentioning

confidence: 99%

The Fault-Tolerant Metric Dimension of Cographs

Vietz

Wanke

2019

Fundamentals of Computation Theory

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A vertex set U ⊆ V of an undirected graph G = (V, E) is a resolving set for G if for every two distinct vertices u, v ∈ V there is a vertex w ∈ U such that the distance between u and w and the distance between v and w are different. A resolving set U is fault-tolerant if for every vertex u ∈ U set U \ {u} is still a resolving set. The (fault-tolerant) Metric Dimension of G is the size of a smallest (fault-tolerant) resolving set for G. The weighted (fault-tolerant) Metric Dimension for a given cost function c : V −→ R+ is the minimum weight of all (fault-tolerant) resolving sets. Deciding whether a given graph G has (fault-tolerant) Metric Dimension at most k for some integer k is known to be NPcomplete. The weighted fault-tolerant Metric Dimension problem has not been studied extensively so far. In this paper we show that the weighted fault-tolerant metric dimension problem can be solved in linear time on cographs.

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“…They showed that for the standard parameter the problem is W [2]-complete on general graphs, even for those with maximum degree at most three [13]. Foucaud et al showed that for interval graphs the problem is FPT for the standard parameter [9] [10]. Afterwards Belmonte et al extended this result to the class of graphs with bounded treelength, which is a superclass of interval graphs and also includes chordal, permutation and ATfree graphs [1].…”

Section: Introductionmentioning

confidence: 99%

Computing the Metric Dimension by Decomposing Graphs into Extended Biconnected Components

Vietz

Hoffmann

Wanke

2018

WALCOM: Algorithms and Computation

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A vertex set U ⊆ V of an undirected graph G = (V, E) is a resolving set for G, if for every two distinct vertices u, v ∈ V there is a vertex w ∈ U such that the distances between u and w and the distance between v and w are different. The Metric Dimension of G is the size of a smallest resolving set for G. Deciding whether a given graph G has Metric Dimension at most k for some integer k is well-known to be NP-complete. Many research has been done to understand the complexity of this problem on restricted graph classes. In this paper, we decompose a graph into its so called extended biconnected components and present an efficient algorithm for computing the metric dimension for a class of graphs having a minimum resolving set with a bounded number of vertices in every extended biconnected component. Further we show that the decision problem Metric Dimension remains NP-complete when the above limitation is extended to usual biconnected components.

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Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

Cited by 13 publications

References 29 publications

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

The Fault-Tolerant Metric Dimension of Cographs

Computing the Metric Dimension by Decomposing Graphs into Extended Biconnected Components

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