• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1085-1100
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• 2008

A set D of vertices of a graph G = (V(G),E(G)) is called a dominating set if every vertex of V(G) - D is adjacent to at least one element of D. The domination number of G, denoted by ${\gamma}(G)$, is the size of its smallest dominating set. Haynes et al.[5] present a conjecture: For any graph G with ${\delta}(G){\geq}k$,$\gamma(G){\leq}\frac{k}{3k-1}n$. When $k\;{\neq}\;6$, the conjecture was proved in [7], [8], [10], [12] and [13] respectively. In this paper we prove that every graph G on n vertices with ${\delta}(G)\;{\geq}\;6$ has a dominating set of order at most $\frac{6}{17}n$. Thus the conjecture was completely proved.

• 대한수학회보
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• 제59권1호
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• pp.167-177
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• 2022

A vertex v of a graph G = (V, E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is called a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The minimum cardinality of a double vertex-edge dominating set of G is the double vertex-edge domination number γ_dve(G). In this paper, we provide an upper bound on the double vertex-edge domination number of trees in terms of the order n, the number of leaves and support vertices, and we characterize the trees attaining the upper bound. Finally, we design a polynomial time algorithm for computing the value of γ_dve(T) for any trees. This gives an answer of an open problem posed in [4].

• Kyungpook Mathematical Journal
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• 제62권1호
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• pp.193-204
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• 2022

A subset S of V(G), where G is a simple undirected graph, is a hop dominating set if for each v ∈ V(G)＼S, there exists w ∈ S such that d_G(v, w) = 2 and it is a locating-hop set if N_G(v, 2) ∩ S ≠ N_G(v, 2) ∩ S for any two distinct vertices u, v ∈ V(G)＼S. A set S ⊆ V(G) is a locating-hop dominating set if it is both a locating-hop and a hop dominating set of G. The minimum cardinality of a locating-hop dominating set of G, denoted by 𝛄_lh(G), is called the locating-hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the locating-hop dominating sets in graphs under some binary operations.

• Journal of applied mathematics & informatics
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• 제41권4호
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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.

• Journal of applied mathematics & informatics
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• 제41권6호
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• pp.1193-1207
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• 2023

In this paper, we introduce an iP₂ extension of a star graph S_n for n ≥ 2 and 1 ≤ i ≤ n - 1. Certain general properties satisfied by order, size, domination (or Roman) numbers γ (or γ_R) of an iP₂ extended star graph are studied. Finally, we study how the parameters such as chromatic number and harmonious chromatic number are affected when an iP₂ extension process acts on the star graphs.

• 한국산업안전학회:학술대회논문집
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• 한국안전학회 1998년도 추계 학술논문발표회 논문집
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• pp.231-237
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• 1998

현대산업에서는 눈부신 기술의 발달로 모든 시스템이 대형화, 복잡화되고 있다. 이들 시스템의 고장은 사회적으로 커다란 문제를 야기할 수 있으며, 이에 이들 시스템의 신뢰도 계산은 필요하다. 이때 신뢰도계산을 목적으로 Network나 통신망 등의 시스템구조는 비방향성 graph로 아주 쉽게 표현될 수 있다. 지금까지는 시스템의 신뢰도 계산방법중 가장 빠른 것으로는 domination이론을 이용한 것들이 발표되었으나, 모두 방향성 graph(directed graph)에 대한 연구(1∼8)이었다. 이에 본 연구에서는 non direct graph에서 domination성질을 관찰하여보고, 이의 결과를 기초로 하여 효과적인 신뢰도 계산방법과 식을 제시하여 본다. (중략)

• 대한수학회보
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• 제33권3호
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• pp.335-342
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• 1996

Suppose $k$ is a field and $M$ is the set of all $m \times n$ matrices over $k$. If T is a linear operator on $M$ and f is a function defined on $M$, then T preserves f if f(T(A)) = f(A) for all $A \in M$.

• Communications for Statistical Applications and Methods
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• 제20권4호
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• pp.329-337
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• 2013

Consider a p-variate($p{\geq}3$) normal distribution with mean ${\theta}$ and covariance matrix ${\sum}={\sigma}^2{\mathbf{I}}_p$ for any unknown scalar ${\sigma}^2$. In this paper we improve the James-Stein estimator of ${\theta}$ in cases of shrinking toward some vectors using the Stein variance estimator. It is also shown that this domination does not hold for the positive part versions of these estimators.

• 대한수학회보
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• 제54권3호
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• pp.917-933
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• 2017

In this paper, seven equivalent conditions of complete moment convergence and complete integral convergence for negatively orthant dependent (NOD, in short) sequences are shown under two cases: identical distribution and stochastic domination. The results obtained in the paper improve and generalize the corresponding ones of Liang et al. [10]). In addition, an extension of the Baum-Katz complete convergence theorem: six equivalent conditions of complete convergence is established.

• 대한수학회논문집
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• 제28권4호
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• pp.857-863
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• 2013

In this paper, we prove that the total domination number of a $P_6$-free graph of order $n{\geq}3$ and minimum degree at least one which is not the cycle of length 6 is at most $\frac{n+1}{2}$, and the bound is sharp.