• Journal of the Korean Mathematical Society
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• v.48 no.3
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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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• 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 $d_{rk}(D)$.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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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.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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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 $0 {\leq}t{\leq}2$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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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.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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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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• v.61 no.3
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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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• v.24 no.2
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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.

• Journal of KIISE:Information Networking
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• v.34 no.3
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• pp.226-238
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• 2007

In this paper, we present a novel fully distributed topology control protocol that can construct the multiple-ring topology of Minimal Connected Dominating Set (MCDS) as the transport backbone for mobile ad hoc networks. It makes a topology from the minimal nodes that are chosen from all the nodes, and the constructed topology is comprised of the minimal physical links while preserving connectivity. This topology reduces the interference. The all nodes work as the nodes of the distributed parallel Boltzmann machine, of which the objective function is consisted of two Boltzmann factors: the link degree and the connection domination degree. To define these Boltzmann factors, we extend the Connected Dominating Set into a fuzzy set, and also define the fuzzy set of nodes by which the multiple-ring topology can be constructed. To construct the transport backbone of the mobile ad hoc network, the proposed protocol chooses the nodes that are the strong members of these two fuzzy sets as the clusterheads. We also ran simulations to provide the quantitative comparison against the related works in terms of the packet loss rate and the energy consumption rate. As a result, we show that the network that is constructed by the proposed protocol has far better than the other ones with respect to the packet loss rate and the energy consumption rate.