Domination chain: Characterisation, classical complexity, parameterised complexity and approximability

Bazgan, Cristina; Branković, Ljiljana; Casel, Katrin; Fernau, Henning

doi:10.1016/j.dam.2019.10.005

Cited by 10 publications

(7 citation statements)

References 64 publications

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“…This problem is also one of the six problems of the well-known domination chain (see [2,18]) and is somewhat one which has fewer results, compared to the famous Dominating Set and Independent Set problems. Increasing our understanding of the Upper Dominating Set problem compared to these two famous problems is another motivation.…”

Section: Introductionmentioning

confidence: 99%

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

Dublois¹,

Lampis²,

Paschos³

2021

Preprint

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An upper dominating set is a minimal dominating set in a graph. In the Upper Dominating Set problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for Upper Dominating Set, as well as its sub-exponential approximation. First, we prove that, under ETH, k-Upper Dominating Set cannot be solved in time)), and in the same time we show under the same complexity assumption that for any constant ratio r and any ε > 0, there is no r-approximation algorithm running in time O(n k 1−ε ). Then, we settle the problem's complexity parameterized by pathwidth by giving an algorithm running in time O * (6 pw ) (improving the current best O * (7 pw )), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an r-approximation in time n O(n/r) , for any desired approximation ratio r < n. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio r > 1 and ε > 0, no algorithm can output an r-approximation in time n (n/r) 1−ε . Hence, we completely characterize the approximability of the problem in sub-exponential time.

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Section: Introductionmentioning

confidence: 99%

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

Dublois¹,

Lampis²,

Paschos³

2021

Preprint

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“…In this spirit, one could ask similar decision questions for other parameters of the famous domination chain, as introduced in [13]. For (a survey of) more recent computational results, we refer to [6,7]. As it is known that all these parameters can di er arbitrarily, the approximation questions discussed in the previous paragraph can be asked analogously for any pair of parameters of the domination chain, or similarly for other graph parameters where inequality relations are known.…”

Section: Discussionmentioning

confidence: 99%

Recognizing well-dominated graphs is coNP-complete

Agrawal¹,

Fernau²,

Kindermann³

et al. 2022

Preprint

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A graph G is well-covered if every minimal vertex cover of G is minimum, and a graph G is well-dominated if every minimal dominating set of G is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were rst introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs was proved coNP-complete by [Chvátal and Slater, AODM 1993] and by [Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing well-dominated graphs has been left open since their introduction. We close this complexity gap by proving that recognizing well-dominated graphs is coNP-complete. This solves a well-known open question (c.f. [Levit and Tankus, DM 2017] and [Gözüpek, Hujdurovic and Milanič, DMTCS 2017]), which was rst asked in [Caro, Sebő and Tarsi, JAlg 1996]. Surprisingly, our proof is quite simple, although it was a long-standing open problem. Finally, we show that recognizing well-totally-dominated graphs is coNP-complete, answering a question of [Bahadır, Ekim, and Gözüpek, AMC 2021].

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“…The domination chain proposed by Cockayne et al is mathematically profound and aesthetically appealing (see Section 5). A recent survey by Bazgan et al lists the most important results regarding the domination chain parameters [2]. In this paper, we partially address the problem raised by Case et al in [3].…”

Section: Introductionmentioning

confidence: 99%

Independent partial domination

Nithya

Kureethara

2021

Cubo

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For p ∈ (0, 1], a set S ⊆ V is said to p-dominate or partially dominate a graphThe minimum cardinality among all p-dominating sets is called the p-domination number and it is denoted by γp(G). Analogously, the independent partial domination (ip(G)) is introduced and studied here independently and in relation with the classical domination. Further, the partial independent set and the partial independence number βp(G) are defined and some of their properties are presented. Finally, the partial domination chain is established as γpRESUMEN Para p ∈ (0, 1], un conjunto S ⊆ V se dice que pdomina o parcialmente domina un grafo G = (V, E) si |N [S]| |V |

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Domination chain: Characterisation, classical complexity, parameterised complexity and approximability

Cited by 10 publications

References 64 publications

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

Recognizing well-dominated graphs is coNP-complete

Independent partial domination

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