A linear time algorithm for metric dimension of cactus block graphs

Hoffmann, Stefan; Elterman, Alina; Wanke, Egon

doi:10.1016/j.tcs.2016.03.024

Cited by 17 publications

(9 citation statements)

References 10 publications

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“…There are several algorithms for computing a minimum resolving set in polynomial time for such classes, for example trees [CEJO00,KRR96], wheels [HMP + 05], grid graphs [MT84], k-regular bipartite graphs [BBS + 11], amalgamation of cycles [IBSS10], cactus block graphs [HEW16], graphs with size-constrained biconnected components [VHW19] and outerplanar graphs [DPSL12]. The approxibility of the metric dimension has been studied for bounded degree, dense and general graphs in [HSV12].…”

Section: Strong Metric Dimensionmentioning

confidence: 99%

On the Strong Metric Dimension of directed co-graphs

Schmitz¹,

Wanke²

2021

Preprint

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Let G be a strongly connected directed graph and u, v, w ∈ V (G) be three vertices. Then w strongly resolves u to v if there is a shortest u-w-path containing v or a shortest w-v-path containing u. A set R ⊆ V (G) of vertices is a strong resolving set for a directed graph G if for every pair of vertices u, v ∈ V (G) there is at least one vertex in R that strongly resolves u to v and at least one vertex in R that strongly resolves v to u. The distances of the vertices of G to and from the vertices of a strong resolving set R uniquely define the connectivity structure of the graph. The Strong Metric Dimension of a directed graph G is the size of a smallest strong resolving set for G. The decision problem Strong Metric Dimension is the question whether G has a strong resolving set of size at most r, for a given directed graph G and a given number r. In this paper we study undirected and directed co-graphs and introduce linear time algorithms for Strong Metric Dimension. These algorithms can also compute strong resolving sets for co-graphs in linear time.

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Section: Strong Metric Dimensionmentioning

confidence: 99%

On the Strong Metric Dimension of directed co-graphs

Schmitz¹,

Wanke²

2021

Preprint

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“…Epstein et al [9] show that Metric DiMension (and even its vertex-weighted variant) can be solved in polynomial time on cographs and forests augmented by a constant number of edges. Hoffmann et al [16] obtain a linear algorithm on cactus block graphs.…”

Section: Introductionmentioning

confidence: 99%

Metric Dimension Parameterized By Treewidth

Bonnet

Purohit

2021

Algorithmica

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A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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“…If distinct vertices of G have distinct representation with respect to W , then W is called resolving set for G. A resolving set of minimum cardinality is a metric basis for G, and its cardinality is called the metric dimension of G, denoted by dim(G). For more detail on metric dimension, we refer to [5,9,20]. Metric dimension is an essential tool in image processing and pattern recognition (see [14]).…”

Section: Introductionmentioning

confidence: 99%

Edge metric dimension of some classes of circulant graphs

Ahsan

Zahid

Zafar

2020

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

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A linear time algorithm for metric dimension of cactus block graphs

Cited by 17 publications

References 10 publications

On the Strong Metric Dimension of directed co-graphs

On the Strong Metric Dimension of directed co-graphs

Metric Dimension Parameterized By Treewidth

Edge metric dimension of some classes of circulant graphs

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