• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.515-523
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• 2019

Consider a graph G of order n and maximum degree ${\Delta}$. Let $f:V(G){\rightarrow}\{0,1,{\cdots},{\lceil}{\frac{{\Delta}}{2}}{\rceil}+1\}$ be a function that labels the vertices of G. Let $B_0=\{v{\in}V(G):f(v)=0\}$. The function f is a strong Roman dominating function for G if every $v{\in}B_0$ has a neighbor w such that $f(w){\geq}1+{\lceil}{\frac{1}{2}}{\mid}N(w){\cap}B_0{\mid}{\rceil}$. In this paper, we study the bounds on strong Roman domination numbers of the Cartesian product $P_m{\square}P_k$ of paths $P_m$ and paths $P_k$. We compute the exact values for the strong Roman domination number of the Cartesian product $P_2{\square}P_k$ and $P_3{\square}P_k$. We also show that the strong Roman domination number of the Cartesian product $P_4{\square}P_k$ is between ${\lceil}{\frac{1}{3}}(8k-{\lfloor}{\frac{k}{8}}{\rfloor}+1){\rceil}$ and ${\lceil}{\frac{8k}{3}}{\rceil}$ for $k{\geq}8$, and that both bounds are sharp bounds.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.22 no.8
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• pp.1114-1122
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• 2018

In this paper, we propose a theoretical framework for provisioning marginal resources in wired and wireless computer networks which include Internet. In more detail, we propose a mathematical model for the upper bounds of marginal capacity in grid networks, where the resource is designed a priori by normal traffic estimation and marginal resource is prepared for unexpected events such as natural disasters and abrupt flash crowd in public affairs. To be specific, we propose a method to evaluate an upper bound for minimum marginal capacity for an arbitrary grid topology using the concept of a strong Roman domination number. To that purpose, we introduce a graph theory to model and analyze the characteristics of general grid structure networks. After that we propose a new tight upper bound for the strong Roman domination number. Via a numerical example, we show the validity of the proposition.

• 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 applied mathematics & informatics
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• v.41 no.6
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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.

• 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.

• Journal of the Korea Fashion and Costume Design Association
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• v.14 no.4
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• pp.29-42
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• 2012

Historically, the pattern and technology of Copts' textiles, who were a minority in Egypt, have been studied a lot in the textile sector due to its unique characteristics. Unlike ones in other regions that appeared around the same time, the overall configuration ratio of the patterns looks exaggerated or distorted because they expressed it plainly by interpreting the world with ordinary people's eyes. Also, because it had used mixed linen and woolen yarns, harsh expression way and the use of various colors have been one of the features in Coptic textiles. Coptic textiles, which have been developed along with the historical development of continued domination from neighboring countries, have expressed the effects of the Roman Empire, Christ, Christianity, and Islam on the pattern of its fabric. This study analyzed its characteristics which make people smile by the way of expressing a simple and humorous representation of the textiles and categorized them as Humoristic Beauty - the aesthetic category of humorous feature. In this study, the Humoristic Beauty in Coptic textiles has been analyzed in terms of the following three smiles; the smiles coincidental with the flow of time, the smiles made by the shaping of distorted proportions and appearance, and the smiles like folk-paintings made by a rustic expression way. This study shows the possibility of the further studies on the textile patterns history from a different angle. I look forward to more detailed analysis in the follow-up studies in the future.