• East Asian mathematical journal
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• v.31 no.5
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• pp.717-725
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• 2015

Velzen introduced the rigid and relaxed dominating set games and showed that the rigid game being balanced is equivalent to the relaxed game being balanced in 2004. After then various variants of dominating set games were introduced and it was shown that for each variant, a rigid game being balanced is equivalent to a relaxed game being balanced. It is natural to ask if for any other variant of dominating set game, the balancedness of a rigid game and the balancedness of a relaxed game are equivalent. In this paper, we show that the answer for the question is negative by considering the rigid and relaxed paired-domination games, which is considered as a variant of dominating set games. We characterize the cores of both games and show that the rigid game being balanced is not equivalent to the relaxed game being balanced. In addition, we study the cores of paired-dominations games on paths and cycles.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.10 no.4
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• pp.1661-1678
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• 2016

Clustering is one of the key technologies in vehicular networks. Constructing and maintaining stable clusters is a challenging task in high mobility environments. DWCM (Distributed and Weighted Clustering based on Mobility Metrics) is proposed in this paper based on the d-hop dominating set of the network. Each vehicle is assigned a priority that describes the cluster relationship. The cluster structure is determined according to the d-hop dominating set, where the vehicles in the d-hop dominating set act as the cluster head nodes. In addition, cluster maintenance handles the cluster structure changes caused by node mobility. The rationality of the proposed algorithm is proven. Simulation results in the NS-2 and VanetMobiSim integrated environment demonstrate the performance advantages.

• Journal of information and communication convergence engineering
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• v.9 no.1
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• pp.105-109
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• 2011

In this paper, we study the dominating-free sest which is defined as follows: k points called servers and n points called clients in the plane are given. For a point p in the plane is said to be dominated by a client c if for every server s, the distance between s and p is greater than the distance between s and c. The dominating-free set is the set of points in the plane which aren't dominated by any client. We present an O(nklogk+$n^2k$) time algorithm for computing the dominating-free set under the $L_1$-metric. Specially, we present an O(nlogn) time algorithm for the problem when k=2. The algorithm uses some variables and 1-dimensional arrays as its principle data structures, so it is easy to implement and runs fast.

• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.741-752
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• 2023

A new variant of vertex edge domination, namely semi total vertex edge domination has been introduced in the present paper. A subset S of the vertex set V of a graph G is said to be a semi total vertex edge dominating set(stved - set), if it is a vertex edge dominating set of G and each vertex in S is within a distance two of another vertex in S. An stved-set of G having minimum cardinality is said to be an γ_stve(G)- set and its cardinality is denoted by γ_stve(G). Bounds for γ_stve(G) - set have been given in terms of various graph theoretic parameters and graphs attaining the bounds have been characterized. In particular, bounds for trees have been obtained and extremal trees have been characterized.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.1-12
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• 2021

A dominating set S of a graph G is said to be nonsplit dominating set if the subgraph ⟨V - S⟩ is connected. The minimum cardinality of a nonsplit dominating set is called the nonsplit domination number and is denoted by ��ns(G). For a minimum nonsplit dominating set S of G, a set T ⊆ S is called a forcing subset for S if S is the unique ��ns-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing nonsplit domination number of S, denoted by f��ns(S), is the cardinality of a minimum forcing subset of S. The forcing nonsplit domination number of G, denoted by f��ns(G) is defined by f��ns(G) = min{f��ns(S)}, where the minimum is taken over all ��ns-sets S in G. The forcing nonsplit domination number of certain standard graphs are determined. It is shown that, for every pair of positive integers a and b with 0 ≤ a ≤ b and b ≥ 1, there exists a connected graph G such that f��ns(G) = a and ��ns(G) = b. It is shown that, for every integer a ≥ 0, there exists a connected graph G with f��(G) = f��ns(G) = a, where f��(G) is the forcing domination number of the graph. Also, it is shown that, for every pair a, b of integers with a ≥ 0 and b ≥ 0 there exists a connected graph G such that f��(G) = a and f��ns(G) = b.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.181-192
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• 2003

We count the numbers of independent dominating sets of rooted labeled trees, ordinary labeled trees, and recursive trees, respectively.

• Journal of KIISE:Computer Systems and Theory
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• v.27 no.6
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• pp.608-618
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• 2000

We consider the problem of placing resources in a distributed computing system so that certain performance requirements may be met while minimizing the number of required resource copies, irrespective of node or link failures. To meet the requirements for high performance and high availability, minimum number of resource copies should be placed in such a way that each node has at least two copies on the node or its neighbor nodes. This is called the fault-tolerant resource placement problem in this paper. The structure of a parallel or a distributed computing system is represented by a graph. The fault-tolerant placement problem is first transformed into the problem of finding the smallest fault-tolerant dominating set in a graph. The dominating set problem is known to be NP-complete. In this paper, searching for the smallest fault-tolerant dominating set is formulated as a state-space search problem, which is then solved optimally with the well-known A* algorithm. To speed up the search, we derive heuristic information by analyzing the properties of fault-tolerant dominating sets. Some experimental results on various regular and random graphs show that the search time can be reduced dramatically using the heuristic information.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.505-519
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• 2023

Let G = (V, E) be a graph. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality taken over all dominating sets of G. A dominating set S is called a connected dominating set if the subgraph induced by S is connected. The minimum cardinality taken over all connected dominating sets of G is called the connected domination number of G, and is denoted by γ_c(G). In this paper, we investigate the Nordhaus-Gaddum type results for the connected domination number and its derived graphs like line graph, subdivision graph, power graph, block graph and total graph, and characterize the extremal graphs.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.69-81
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• 2016

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set $\{1,2,{\ldots},k\}$ such that for any vertex $v{\in}V(D)$ with $f(v)={\emptyset}$ the condition ${\cup}_{u{\in}N^-(v)}$ $f(u)=\{1,2,{\ldots},k\}$ is fulfilled, where $N^-(v)$ is the set of in-neighbors of v. A set $\{f_1,f_2,{\ldots},f_d\}$ of k-rainbow dominating functions on D with the property that $\sum_{i=1}^{d}{\mid}f_i(v){\mid}{\leq}k$ for each $v{\in}V(D)$, is called a k-rainbow dominating family (of functions) on D. The maximum number of functions in a k-rainbow dominating family on D is the k-rainbow domatic number of D, denoted by $d_{rk}(D)$. In this paper we initiate the study of the k-rainbow domatic number in digraphs, and we present some bounds for $d_{rk}(D)$.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.12A
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• pp.1188-1197
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• 2010

A most important thing in a connected dominating set(CDS)-based routing in a wireless ad-hoc network is to select a minimum number of dominating nodes and then build a backbone network which is made of them. Node failure in a CDS is an event of non-negligible probability. For applications where fault tolerance is critical, a traditional dominating-set based routing may not be a desirable form of clustering. It is necessary to minimize the frequency of reconstruction of a CDS to reduce message overhead due to message flooding. The idea is that by finding alternative nodes within a restricted range and locally reconstructing a CDS to include them, instead of totally reconstructing a new CDS. With the proposed algorithm, the resulting number of dominating nodes after partial reconstruction of CDS is not changed and also its execution time is faster than well-known algorithm of construction of CDS by 20~40%. In the case of high mobility situation, the proposed algorithm gives better results for the performance metrics, packet receive ratio and energy consumption.