• 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.

• 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.

• Kyungpook Mathematical Journal
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• 제59권3호
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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.

• 한국정보통신학회논문지
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• 제22권8호
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• pp.1114-1122
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• 2018

본 연구에서는 그리드 네트워크에 대해서 정상적인 트래픽 예측에 의해서 설계된 자원 이외에 예측 불가능한 비상사태를 대비하기 위한 추가자원의 한계용량 설계에 관한 해석적 모델을 제안한다. 구체적으로는 그리드 네트워크 전체를 대상으로 한계용량을 산정하는 경우의 최솟값에 대한 상한을 예측하는 방법을 제안한다. 이를 위해서 본 논문에서는 그리드 네트워크를 그리드 그래프로 추상화하여 Roman domination number의 개념을 이용한 해석적 기법을 통해서 한계용량의 상한을 도출한다.

• 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 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.