• 대한수학회지
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• 제48권3호
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• pp.551-563
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• 2011

Let ${\kappa}$ be a positive integer and let G be a simple graph with vertex set V(G). A function f : V (G) ${\rightarrow}$ {-1, 1} is called a signed total ${\kappa}$-dominating function if ${\sum}_{u{\in}N({\upsilon})}f(u){\geq}{\kappa}$ for each vertex ${\upsilon}{\in}V(G)$. A set ${f_1,f_2,{\ldots},f_d}$ of signed total ${\kappa}$-dominating functions of G with the property that ${\sum}^d_{i=1}f_i({\upsilon}){\leq}1$ for each ${\upsilon}{\in}V(G)$, is called a signed total ${\kappa}$-dominating family (of functions) of G. The maximum number of functions in a signed total ${\kappa}$-dominating family of G is the signed total k-domatic number of G, denoted by $d^t_{kS}$(G). In this note we initiate the study of the signed total k-domatic numbers of graphs and present some sharp upper bounds for this parameter. We also determine the signed total signed total ${\kappa}$-domatic numbers of complete graphs and complete bipartite graphs.

• Kyungpook Mathematical Journal
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• 제56권1호
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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 $d_{rk}(D)$.

• 대한수학회지
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• 제46권6호
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• pp.1309-1318
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• 2009

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) $\rightarrow$ {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices $\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$ with $f(\upsilon_i)$ = 2 for i = 1, 2, $\ldot$, k. The weight of a Roman k-dominating function is the value f(V (G)) = $\sum_{u{\in}v(G)}$ f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number ${\gamma}_{kR}$(G) of G. Note that the Roman 1-domination number $\gamma_{1R}$(G) is the usual Roman domination number $\gamma_R$(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

• 대한수학회보
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• 제56권1호
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• pp.31-44
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• 2019

A graph theoretical model called Roman domination in graphs originates from the historical background that any undefended place (with no legions) of the Roman Empire must be protected by a stronger neighbor place (having two legions). It is applicable to military and commercial decision-making problems. A Roman dominating function for a graph G = (V, E) is a function $f:V{\rightarrow}\{0,1,2\}$ such that every vertex v with f(v)=0 has at least a neighbor w in G for which f(w)=2. The Roman domination number of a graph is the minimum weight ${\sum}_{v{\in}V}\;f(v)$ of a Roman dominating function. In order to deal a problem of a Roman domination-type defensive strategy under multiple simultaneous attacks, ${\acute{A}}lvarez$-Ruiz et al. [1] initiated the study of a new parameter related to Roman dominating function, which is called strong Roman domination. ${\acute{A}}lvarez$-Ruiz et al. posed the following problem: Characterize the graphs G with equal strong Roman domination number and Roman domination number. In this paper, we construct a family of trees. We prove that for a tree, its strong Roman dominance number and Roman dominance number are equal if and only if the tree belongs to this family of trees.

• 대한수학회보
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• 제50권6호
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• pp.2021-2026
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• 2013

Let G = (V,E) be a graph. A function $f:V{\rightarrow}\{-1,+1\}$ defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, ${\gamma}^s_t(G)$, is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen graphs P(n, 2) and prove that for any integer $n{\geq}6$, ${\gamma}^s_t(P(n,2))=2[\frac{n}{3}]+2t$, where $t{\equiv}n(mod\;3)$ and $0 {\leq}t{\leq}2$.

• 대한수학회보
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• 제50권2호
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• pp.469-473
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• 2013

Let G = (V, E) be a graph and k be a positive integer. A $k$-dominating set of G is a subset $S{\subseteq}V$ such that each vertex in $V{\backslash}S$ has at least $k$ neighbors in S. A Roman $k$-dominating function on G is a function $f$ : V ${\rightarrow}$ {0, 1, 2} such that every vertex ${\upsilon}$ with $f({\upsilon})$ = 0 is adjacent to at least $k$ vertices ${\upsilon}_1$, ${\upsilon}_2$, ${\ldots}$, ${\upsilon}_k$ with $f({\upsilon}_i)$ = 2 for $i$ = 1, 2, ${\ldots}$, $k$. In the paper titled "Roman $k$-domination in graphs" (J. Korean Math. Soc. 46 (2009), no. 6, 1309-1318) K. Kammerling and L. Volkmann showed that for any graph G with $n$ vertices, ${{\gamma}_{kR}}(G)+{{\gamma}_{kR}(\bar{G})}{\geq}$ min $\{2n,4k+1\}$, and the equality holds if and only if $n{\leq}2k$ or $k{\geq}2$ and $n=2k+1$ or $k=1$ and G or $\bar{G}$ has a vertex of degree $n$ - 1 and its complement has a vertex of degree $n$ - 2. In this paper we find a counterexample of Kammerling and Volkmann's result and then give a correction to the result.

• 대한수학회보
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• 제43권2호
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• pp.265-275
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• 2006

In this paper, we introduce a fractional domination game arising from fractional domination problems on graphs and focus on its balancedness and concavity. We first characterize the core of the fractional domination game and show that its core is always non-empty taking use of dual theory of linear programming. Furthermore we study concavity of this game.

• Kyungpook Mathematical Journal
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• 제61권3호
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• pp.661-669
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• 2021

For a simple, undirected graph G = (V, E), a perfect Roman dominating function (PRDF) f : V → {0, 1, 2} has the property that, every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a PRDF is the sum f(V) = ∑_v∈V f(v). The minimum weight of a PRDF is called the perfect Roman domination number, denoted by γ_R^P(G). Given a graph G and a positive integer k, the PRDF problem is to check whether G has a perfect Roman dominating function of weight at most k. In this paper, we first investigate the complexity of PRDF problem for some subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. Then we show that PRDF problem is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs.

• Journal of the Korean Statistical Society
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• 제24권2호
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• pp.361-373
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• 1995

Three remarks are offered, pertaining to classes of estimators Pitman-dominating a given estimator. The first remark concerns incorporating general loss in the construction of such classes. The second remark concerns Pitman domination comparisons amongst the members of such classes. The third remark concerns construction of such a class in the location-scale case.

• 한국정보과학회논문지:정보통신
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• 제34권3호
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• pp.226-238
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• 2007

본 논문은 연결 지배 집합에 속하는 노드들로 애드혹 망의 위상을 구성하는 완전 분산형 위상 제어 프로토콜을 제시한다. 제안한 프로토콜은 가능한 최소의 노드 수로 위상을 구성할 수 있게 하여 패킷 전송 시 발생하는 간섭을 줄일 수 있다. 제안한 프로토콜의 알고리즘 복잡도는 O(1)이다. 각 노드는 분산된 병렬 볼츠만 기계의 한 노드로서 동작한다. 이 볼츠만 기계의 목적 함수를 연결의 차수와 연결 지배 정도를 표현하는 두 개의 볼츠만 인수로 구성한다. 이 볼츠만 인수들을 정의하기 위해 두 개의 퍼지 집합을 정의한다. 하나는 연결 지배 노드로 이루어진 퍼지 집합이며, 다른 하나는 다중-링 위상 구성이 가능한 노드로 이루어진 퍼지 집합이다. 제안한 프로토콜은 이 두 퍼지 집합의 강한 원소 노드들을 애드혹 망의 클러스터 헤드로 선택한다. 모의 실험을 통해 패킷 손실율과 에너지 소비율 측면에서 제안 프로토콜이 기존 방법에 비해 우수함을 확인하였다.