Minimal doubly resolving sets of prism graphs

Čangalović, Mirjana; Kratica, Jozef; Kovačevíc-Vujčić, Vera; Stojanović, Milica

doi:10.1080/02331934.2013.772999

Cited by 30 publications

(24 citation statements)

References 6 publications

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“…Furthermore, combining (a)-(e) and Observation 3, we deduce from Theorem F.3 that the minimum weighted DRS of G must be one of S(s, left) with s ∈ α(1), S ′ (s, left) with s ∈ β(1), S(s, right) with s ∈ γ(1), and S ′ (s, right) with s ∈ β(1) ∩ γ(1). These sets satisfy all the conditions in Theorem F. 3. Hence the weight of the optimal DRS is given by 6.…”

Section: Algorithm 3 Finding Minimum Weighted Drs Of General Wheelsmentioning

confidence: 96%

“…Corollary 4. 3. If some minimum weighted DRS has cardinality 3, then there exists a minimum weighted DRS of G that contains vertex v p .…”

Section: K-edge-augmented Treesmentioning

confidence: 99%

“…By examining all vertices in P a , without loss of generality we assume that vertex 1 ∈ V (P a ) is an observer in a minimum weighted DRS. Our aim is to construct an observer set S of minimum weight on the condition that 1 ∈ S and S satisfies conditions (i)-(iv) in Theorem F. 3. Such an S must be a minimum weighted DRS of G as guaranteed by Theorem F. 3.…”

Section: F Wheelsmentioning

confidence: 99%

“…Our aim is to construct an observer set S of minimum weight on the condition that 1 ∈ S and S satisfies conditions (i)-(iv) in Theorem F. 3. Such an S must be a minimum weighted DRS of G as guaranteed by Theorem F. 3. Henceforth we assume 1 ∈ S.…”

Section: F Wheelsmentioning

confidence: 99%

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Approximability of the Minimum Weighted Doubly Resolving Set Problem

Chen

Wang

2014

Lecture Notes in Computer Science

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Locating source of diffusion in networks is crucial for controlling and preventing epidemic risks. It has been studied under various probabilistic models. In this paper, we study source location from a deterministic point of view by modeling it as the minimum weighted doubly resolving set (DRS) problem, which is a strengthening of the well-known metric dimension problem.Let G be a vertex weighted undirected graph on n vertices. A vertex subset S of G is DRS of G if for every pair of vertices u, v in G, there exist x, y ∈ S such that the difference of distances (in terms of number of edges) between u and x, y is not equal to the difference of distances between v and x, y. The minimum weighted DRS problem consists of finding a DRS in G with minimum total weight. We establish Θ(ln n) approximability of the minimum DRS problem on general graphs for both weighted and unweighted versions. This is the first work providing explicit approximation lower and upper bounds for minimum (weighted) DRS problem, which are nearly tight. Moreover, we design first known strongly polynomial time algorithms for the minimum weighted DRS problem on general wheels and trees with additional constant k ≥ 0 edges.

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Section: Algorithm 3 Finding Minimum Weighted Drs Of General Wheelsmentioning

confidence: 96%

“…Corollary 4. 3. If some minimum weighted DRS has cardinality 3, then there exists a minimum weighted DRS of G that contains vertex v p .…”

Section: K-edge-augmented Treesmentioning

confidence: 99%

Section: F Wheelsmentioning

confidence: 99%

Section: F Wheelsmentioning

confidence: 99%

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Approximability of the Minimum Weighted Doubly Resolving Set Problem

Chen

Wang

2014

Lecture Notes in Computer Science

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“…The minimum cardinality of a doubly resolving set for G is represented by ψ(G). In case of some convex prism, hamming and polytopes graphs, the minimal doubly resolving sets have been obtained in [24][25][26] respectively.…”

Section: Suppose That Rmentioning

confidence: 99%