Total Domination in Graphs with Diameter 2

Desormeaux, Wyatt J.; Haynes, Teresa W.; Henning, Michael A.; Yeo, Anders

doi:10.1002/jgt.21725

Cited by 14 publications

(8 citation statements)

References 14 publications

(20 reference statements)

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“…The specific case h = 1 and D = 2 has been already studied in the literature since a distance 1-dominating set is a classical dominating set. Desormeaux et al [13] prove that in undirected graphs of diameter 2 the smallest 1-dominating set has size Θ( √ n log n) in the worst case. 7 We provide several lower bounds, with focus on the coefficients λ and δ that relate h and D as h = λ(D + δ).…”

Section: Lower Bounds On H-dominating Setsmentioning

confidence: 99%

See 1 more Smart Citation

New Bounds for Approximating Extremal Distances in Undirected Graphs

Cairo¹,

Grossi²,

Rizzi³

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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This Paper Appears in\ud Cover Image\ud Title Information\ud Published: 2016\ud eISBN: 978-1-61197-433-1\ud Book Code: PRDA16\ud Series: Proceedings\ud Pages: 14\ud Abstract\ud We provide new bounds for the approximation of extremal distances (the diameter, the radius, and the eccentricities of all nodes) of an undirected graph with n nodes and m edges. First, we show under the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [JCSS01] that it is impossible to get a (3/2 – ∊)-approximation of the diameter or a (5/3 – ∊)-approximation of all the eccentricities in O(m2–δ) time for any ∊, δ > 0, even allowing for a constant additive term in the approximation. Second, we present an algorithmic scheme that gives a (2 – 1/2k)-approximation of the diameter and the radius and a (3 – 4/(2k + 1))-approximation of all eccentricities in expected time for any k ≥ 0. For k ≥ 2, this gives a family of previously unknown bounds, and approaches near-linear running time as k grows. Third, we observe a connection between the approximation of the diameter and the h-dominating sets, which are subsets of nodes at distance ≤ h from every other node. We give bounds for the size of these sets, related with the diameter

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Section: Lower Bounds On H-dominating Setsmentioning

confidence: 99%

“…In fact, here we discover that X is an h 0 -dominating set, but in general it can be h 0 < h 7. Desormeaux et al[13] consider total dominating sets. On graphs without isolated nodes, a dominating set of size t can be transformed into a total dominating set of size 2t[19], hence their Ω( √ n log n) lower bound still holds.…”

mentioning

confidence: 99%

New Bounds for Approximating Extremal Distances in Undirected Graphs

Cairo¹,

Grossi²,

Rizzi³

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…First note that the domination number of a diameter two graph may be as large as C √ n log n, see [7]. Given a graph G, and a vertex v, we say that a cop controls v if the cop is on v or on an adjacent vertex.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Cops and Robbers on diameter two graphs

Wagner

2015

Discrete Mathematics

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Abstract. In this short paper we study the game of Cops and Robbers, played on the vertices of some fixed graph G of order n. The minimum number of cops required to capture a robber is called the cop number of G. We show that the cop number of graphs of diameter 2 is at most √ 2n, improving a recent result of Lu and Peng by a constant factor. We conjecture that this bound is still not optimal, and obtain some partial results towards the optimal bound.

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“…The size of a minimum dominating set in graphs of diameter bounded by 2 (hence 2-clubs) has been considered in [8], where the following result is proven.…”

Section: Lemmamentioning

confidence: 99%

Covering a Graph with Clubs

Dondi¹,

Mauri²,

Sikora³

et al. 2019

JGAA

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Finding cohesive subgraphs in a network has been investigatSeveral alternative formulations of cohesive subgraph have been proposed, a notable one of them is s-club, which is a subgraph whose diameter is at most s. Here we consider a natral variant of the well-known Minimum Clique Cover problem, where we aim to cover a given graph with the minimum number of s-clubs, instead of cliques. We study the computational and approximation complexity of this problem, when s is equal to 2 or 3. We show that deciding if there exists a cover of a graph with three 2-clubs is NP-complete, and that deciding if there exists a cover of a graph with two 3-clubs is NP-complete. Then, we consider the approximation complexity of covering a graph with the minimum number of 2-clubs and 3-clubs. We show that, given a graph G = (V, E) to be covered, covering G with the minimum number of 2-clubs is not approximable within factor O(|V | 1/2−ε), for any ε > 0, and covering G with the minimum number of 3-clubs is not approximable within factor O(|V | 1−ε), for any ε > 0. On the positive side, we give an approximation algorithm of factor 2|V | 1/2 log 3/2 |V | for covering a graph with the minimum number of 2-clubs.

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Total Domination in Graphs with Diameter 2

Cited by 14 publications

References 14 publications

New Bounds for Approximating Extremal Distances in Undirected Graphs

New Bounds for Approximating Extremal Distances in Undirected Graphs

Cops and Robbers on diameter two graphs

Covering a Graph with Clubs

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