• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.829-836
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• 2021

In this paper, we consider several domination parameters like perfect domination number, locating-domination number, open-locatingdomination number, etc. in the Mycielski graph M(G) of a graph G. We found upper bounds for locating-domination number of M(G) and computational formulae for perfect locating-domination number and open locating-domination number of M(G). We also showed that the perfect domination number of M(G) is at least that of G plus 1 and that for each positive integer n, there exists a graph G_n such that the perfect domination number of M(G_n) is equal to that of G_n plus n.

• Korean Journal of Mathematics
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• v.7 no.1
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• pp.37-44
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• 1999

We study the relations among the domination number, the independent domination number, and the independence number of an oriented tree and establish their bounds. We also do the same for a binary tree.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.1-12
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• 2021

A dominating set S of a graph G is said to be nonsplit dominating set if the subgraph ⟨V - S⟩ is connected. The minimum cardinality of a nonsplit dominating set is called the nonsplit domination number and is denoted by ��ns(G). For a minimum nonsplit dominating set S of G, a set T ⊆ S is called a forcing subset for S if S is the unique ��ns-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing nonsplit domination number of S, denoted by f��ns(S), is the cardinality of a minimum forcing subset of S. The forcing nonsplit domination number of G, denoted by f��ns(G) is defined by f��ns(G) = min{f��ns(S)}, where the minimum is taken over all ��ns-sets S in G. The forcing nonsplit domination number of certain standard graphs are determined. It is shown that, for every pair of positive integers a and b with 0 ≤ a ≤ b and b ≥ 1, there exists a connected graph G such that f��ns(G) = a and ��ns(G) = b. It is shown that, for every integer a ≥ 0, there exists a connected graph G with f��(G) = f��ns(G) = a, where f��(G) is the forcing domination number of the graph. Also, it is shown that, for every pair a, b of integers with a ≥ 0 and b ≥ 0 there exists a connected graph G such that f��(G) = a and f��ns(G) = b.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1409-1418
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• 2021

In this paper, we study domination in the zero-divisor graph of a *-ring. We first determine the domination number, the total domination number, and the connected domination number for the zero-divisor graph of the product of two *-rings with componentwise involution. Then, we study domination in the zero-divisor graph of a Rickart *-ring and relate it with the clique of the zero-divisor graph of a Rickart *-ring.

• 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.35 no.4
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• pp.843-853
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• 1998

We establish bounds for the domination number of a digraph in terms of the minimum indegree and the order, and then we find a sharp upper bound for the domination number of a weak digraph with minimum indegree one. We also determine the domination number of a random digraph.

• Korean Journal of Mathematics
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• v.9 no.1
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• pp.21-28
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• 2001

We find bounds for the domination number of a tournament and investigate the sharpness of these bounds. We also find the domination number of a random tournament.

• Communications of the Korean Mathematical Society
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• v.27 no.4
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• pp.843-849
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• 2012

Let G = (V, E) be a graph with chromatic number ${\chi}(G)$. dominating set D of G is called a chromatic transversal dominating set (ctd-set) if D intersects every color class of every ${\chi}$-partition of G. The minimum cardinality of a ctd-set of G is called the chromatic transversal domination number of G and is denoted by ${\gamma}_{ct}$(G). In this paper we characterize the class of trees, unicyclic graphs and cubic graphs for which the chromatic transversal domination number is equal to the connected domination number.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.245-250
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• 2020

Let γt(G) and τ(G) denote the total domination number and vertex cover number of graph G, respectively. In this paper, we study the total domination number of the central tree C(T) for a tree T. First, a relationship between the total domination number of C(T) and the vertex cover number of tree T is discussed. We characterize the central trees with equal total domination number and independence number. Applying the first result, we improve the upper bound on the total domination number of C(T) and solve one open problem posed by Kazemnejad et al..

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.1059-1075
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• 2019

Let G be a graph with no isolated vertex. A total dominating set, abbreviated TDS of G is a subset S of vertices of G such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a TDS of G. In this paper, we study the total domination number of central graphs. Indeed, we obtain some tight bounds for the total domination number of a central graph C(G) in terms of some invariants of the graph G. Also we characterize the total domination number of the central graph of some families of graphs such as path graphs, cycle graphs, wheel graphs, complete graphs and complete multipartite graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of central graphs.