Metric Dimension of Bounded Width Graphs

Belmonte, Rémy; Fomin, Fedor V.; Golovach, Petr A.; Ramanujan, M. S.

doi:10.1007/978-3-662-48054-0_10

Cited by 14 publications

(22 citation statements)

References 20 publications

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“…We now generalize our result for trees to graphs with tree decompositions of given width and length. These results also generalize results of [17] for interval graphs and permutation graphs (which have treelength at most 1 and 2, respectively [6]).…”

Section: Using Tree Decompositionssupporting

confidence: 82%

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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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Section: Using Tree Decompositionssupporting

confidence: 82%

“…Now assume that T is a tree of diameter d and metric dimension k attaining the bound. Then equality holds also in (6), in (4)- (5). So the paths P i share no vertices other than central vertices, and each path has length r. Moreover, equality in (5) implies that for the vertices v of P i , the trees T v have order 2, 3, .…”

Section: Treesmentioning

confidence: 96%

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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show abstract

“…Finally, and independently from this work, Belmonte et al [5] proved that tw(G) = O(∆ tl(G) ) for any graph G with maximum degree ∆. They built upon this relation in order to design a fixed-parameter-tractable algorithm to compute the metric dimension on bounded treelength graphs.…”

Section: State Of the Artmentioning

confidence: 96%

“…A partially triangulated (r × r)-grid is any graph that contains an (r × r)-grid as a subgraph and is a subgraph of some planar triangulation of the same (r × r)-grid. A (r, k)-gridoid G is a partially triangulated (r × r)-grid in which k extra edges have been added 5 .…”

Section: 2mentioning

confidence: 99%

To Approximate Treewidth, Use Treelength!

Coudert

Ducoffe

Nisse

2016

SIAM J. Discrete Math.

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International audienceTree-likeness parameters have proven their utility in the design of efficient algorithms on graphs. In this paper, we relate the structural tree-likeness of graphs with their metric tree-likeness. To this end, we establish new upper-bounds on the diameter of minimal separators in graphs. We prove that in any graph G, the diameter of any minimal separator S in G is at most ⌊l(G)/2⌋ · (|S| − 1), with l(G) the length of a longest isometric cycle in G. Our result relies on algebraic methods and on the cycle basis of graphs. We improve our bound for the graphs admitting a distance preserving elimination ordering, for which we prove that any minimal separator S has diameter at most 2 · (|S| − 1). We use our results to prove that the treelength tl(G) of any graph G is at most ⌊l(G)/2⌋ times its treewidth tw(G). In addition, we prove that, for any graph G that excludes an apex graph H as a minor, tw(G) ≤ c_H · tl(G) for some constant c_H only depending on H. We refine this constant when G has bounded genus. Altogether, we obtain a simple O(l(G))-approximation algorithm for computing the treewidth of n-node apex-minor-free graphs in O(n^2)-time

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“…Hence we have the following. 3 Theorem 7 (Hartung and Nichterlein [16,17]). Opt-Detection Pair is NP-hard to approximate within a factor of (1 − ǫ) log n (for any ǫ > 0) for instances of order n, and Detection Pair is W [2]-hard (parameterized by the solution size k).…”

Section: General Approximability and Non-approximabilitymentioning

confidence: 99%

Parameterized and approximation complexity of the detection pair problem in graphs

Foucaud¹,

Klasing²

2017

JGAA

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We study the complexity of the problem Detection Pair. A detection pair of a graph G is a pair (W, L) of sets of detectors with W ⊆ V (G), the watchers, and L ⊆ V (G), the listeners, such that for every pair u, v of vertices that are not dominated by a watcher of W , there is a listener of L whose distances to u and to v are different. The goal is to minimize |W | + |L|. This problem generalizes the two classic problems Dominating Set and Metric Dimension, that correspond to the restrictions L = ∅ and W = ∅, respectively. Detection Pair was recently introduced by Finbow, Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The complexity of monitoring a network with both watchers and listeners. Networks, accepted], who proved it to be NP-complete on trees, a surprising result given that both Dominating Set and Metric Dimension are known to be linear-time solvable on trees. It follows from an existing reduction by Hartung and Nichterlein for Metric Dimension that even on bipartite subcubic graphs of arbitrarily large girth, Detection Pair is NP-hard to approximate within a sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We show, using a reduction to Set Cover, that Detection Pair is approximable within a factor logarithmic in the number of vertices of the input graph. Our two main results are a linear-time 2-approximation algorithm and an FPT algorithm for Detection Pair on trees.

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Metric Dimension of Bounded Width Graphs

Cited by 14 publications

References 20 publications

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

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

To Approximate Treewidth, Use Treelength!

Parameterized and approximation complexity of the detection pair problem in graphs

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