$$k$$ k -Power domination in block graphs

Wang, Chao; Lei, Chen; Lu, Changhong

doi:10.1007/s10878-014-9795-0

Cited by 17 publications

(12 citation statements)

References 22 publications

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“…Cyanoacetamides 327 , previously readily obtained by simply mixing methyl cyanoacetate with primary and secondary amines, could be used as the substrates in the Gewald reaction . Dömling et al.…”

Section: Sulfur As Building Block – Sulfuration Reactionsmentioning

confidence: 99%

Recent Advances in Organic Reactions Involving Elemental Sulfur

2017

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Elemental sulfur hasb een known since Antiquity and found widespread applications in the preparation of black gunpowder,t he synthesis of sulfuric acid as well as other sulfur-containing compounds,a nd the vulcanization of rubber.O ver the last several years,wehave come to better understand its properties andd iscover more applications of elementals ulfuri ns ynthetic organic chemistry.T his review summarizes the advancesf rom 2000i nt he construction of organic molecules using elemental sulfur via sulfuration, oxidation, reduction and redox condensation processes.

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Section: Sulfur As Building Block – Sulfuration Reactionsmentioning

confidence: 99%

Recent Advances in Organic Reactions Involving Elemental Sulfur

2017

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“…Algorithms have been developed for solving the k ‐PDSP on specific graphs. For example, linear time algorithms have been introduced for finding optimal solutions in case of trees (Chang et al, 2012) and block graphs (Liao, 2016; Wang, Chen, & Lu, 2016). It is interesting to note that the minimal k ‐PDS for k > 1 are often of very small sizes; for instance, in Sundara Rajan, Anitha, and Rajasingh (2015) it has been shown that the 2‐PDS is of size 2 for torus, twisted torus, and certain bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

The fixed set search applied to the power dominating set problem

Jovanović

Voß

2020

Expert Systems

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In this article, we focus on solving the power dominating set problem and its connected version. These problems are frequently used for finding optimal placements of phasor measurement units in power systems. We present an improved integer linear program (ILP) for both problems. In addition, a greedy constructive algorithm and a local search are developed. A greedy randomised adaptive search procedure (GRASP) algorithm is created to find near optimal solutions for large scale problem instances. The performance of the GRASP is further enhanced by extending it to the novel fixed set search (FSS) metaheuristic. Our computational results show that the proposed ILP has a significantly lower computational cost than existing ILPs for both versions of the problem. The proposed FSS algorithm manages to find all the optimal solutions that have been acquired using the ILP. In the last group of tests, it is shown that the FSS can significantly outperform the GRASP in both solution quality and computational cost.

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“…Chang et al [6] generalized the power domination to k-power domination by replacing the OR 2 with the following observation rule: If an observed vertex v has at most k unobserved neighbors, then v will send a message to all its unobserved neighbors. The definition also can be found in [21].…”

Section: Introductionmentioning

confidence: 99%

“…The k-power domination number of G, denoted by γ p,k (G), is the minimum cardinality of a k-power dominating set of G. When k = 1, The k-power domination is usual power domination. Both power domination and k-power domination are now well-studied in the literature (see, for example, [1,6,8,9,10,12,21,23,25]).…”

Section: Introductionmentioning

confidence: 99%

Power domination in regular claw-free graphs

Mao

Bing

2020

Discrete Applied Mathematics

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In this paper, we first show that the power domination number of a connected 4-regular claw-free graph on n vertices is at most n+1 5 , and the bound is sharp. The statement partly disprove the conjecture presented by Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a dynamic programming style linear-time algorithm for weighted power domination problem in trees.Let G = (V, E) be a connected, simple graph with vertex set V = V (G) and edgeWe denote K i,j the complete bipartite graph with two partite sets of cardinality i and j, respectively. A claw-free graph is a graph that does not contain a claw, i.e., K 1,3 , as an induced subgraph. We say a subset of V (G) an independent set if no two vertices of the set are adjacent in G. Let x and y are two vertices of G. Denote by d(x, y) the distance of x and y in G. We say a subset of V (G) a packing if no two vertices in the set of distance less than three in G. For two graphs G = (V, E) andthen G and G ′ are disjoint. For any vertex subset X of G, let G − X = G[V \ X] and for X = {x} let G − x = G − {x} for short. For notation and graph theory terminology not defined herein, we in general follow [7]. The original definition of power domination was simplified to the following definition independently in [9, 10, 12, 17] and elsewhere. Definition 1.1 Let G be a graph. A set S ⊆ V (G) is a power dominating set (abbreviated as PDS) of G if and only if all vertices of V (G) have messages either by Observation Rule 1 (abbreviated as OR 1) initially or by Observation Rule 2 (abbreviated as OR 2) recursively.OR 1. A vertex v ∈ S sends a message to itself and all its neighbors. We say that v observes itself and all its neighbors. OR 2. If an observed vertex v has only one unobserved neighbor u, then v will send a message to u. We say that v observes u.Let G = (V, E) be a graph and S be a subset of V . For i ≥ 0, we define the set P i G (S) of vertices observed by S at step i by the following rules:

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$$k$$ k -Power domination in block graphs

Cited by 17 publications

References 22 publications

Recent Advances in Organic Reactions Involving Elemental Sulfur

Recent Advances in Organic Reactions Involving Elemental Sulfur

The fixed set search applied to the power dominating set problem

Power domination in regular claw-free graphs

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