Strong Roman Domination in Grid Graphs

Chen, Xue-Gang,Sohn, Moo Young Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.3

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.

Total domination number of central trees

Xue-gang Chen,손무영,Yu-Feng Wang 대한수학회 2020 대한수학회보 Vol.57 No.1

Let $\gamma_{t}(G)$ and $\tau(G)$ 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..

TREES WITH EQUAL STRONG ROMAN DOMINATION NUMBER AND ROMAN DOMINATION NUMBER

Chen, Xue-Gang,Sohn, Moo Young Korean Mathematical Society 2019 대한수학회보 Vol.56 No.1

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.

Trees with equal strong Roman domination number and Roman domination number

Xue-gang Chen,손무영 대한수학회 2019 대한수학회보 Vol.56 No.1

A graph theoretical model called Roman domination in \linebreak 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 $f : V \to \{0, 1, 2 \}$ 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.~\cite{Ruiz} 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.

Double vertex-edge domination in trees

Xue-gang Chen,손무영 대한수학회 2022 대한수학회보 Vol.59 No.1

A vertex $v$ of a graph $G=(V,E)$ is said to $ve$-dominate every edge incident to $v$, as well as every edge adjacent to these incident edges. A set $S\subseteq V$ is called a double vertex-edge dominating set if every edge of $E$ is $ve$-dominated by at least two vertices of $S$. The minimum cardinality of a double vertex-edge dominating set of $G$ is the double vertex-edge domination number $\gamma_{dve}(G)$. In this paper, we provide an upper bound on the double vertex-edge domination number of trees in terms of the order $n$, the number of leaves and support vertices, and we characterize the trees attaining the upper bound. Finally, we design a polynomial time algorithm for computing the value of $\gamma_{dve}(T)$ for any trees. This gives an answer of an open problem posed in \cite{kri}.

TOTAL DOMINATION NUMBER OF CENTRAL TREES

Chen, Xue-Gang,Sohn, Moo Young,Wang, Yu-Feng Korean Mathematical Society 2020 대한수학회보 Vol.57 No.1

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..

TOTAL DOMINATIONS IN P₆-FREE GRAPHS

Chen, Xue-Gang,Sohn, Moo Young Korean Mathematical Society 2013 대한수학회논문집 Vol.28 No.4

In this paper, we prove that the total domination number of a $P_6$-free graph of order $n{\geq}3$ and minimum degree at least one which is not the cycle of length 6 is at most $\frac{n+1}{2}$, and the bound is sharp.

ON [1, 2]-DOMINATION IN TREES

Chen, Xue-Gang,Sohn, Moo Young Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.2

Chellai et al. [3] gave an upper bound on the [1, 2]-domination number of tree and posed an open question "how to classify trees satisfying the sharp bound?". Yang and Wu [5] gave a partial solution for tree of order n with ${\ell}$-leaves such that every non-leaf vertex has degree at least 4. In this paper, we give a new upper bound on the [1, 2]-domination number of tree which extends the result of Yang and Wu. In addition, we design a polynomial time algorithm for solving the open question. By using this algorithm, we give a characterization on the [1, 2]-domination number for trees of order n with ${\ell}$ leaves satisfying $n-{\ell}$. Thereby, the open question posed by Chellai et al. is solved.

1.54 μm photoluminescence emission at room-temperature of erbium-implanted lithium niobate crystal

Gang Fu,Shi Ling Li,Xue Lin Wang,Feng Chen,Ke Ming Wang 한국물리학회 2008 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.52 No.-

Erbium doped materials are of great interest in integrated optoelectronic technology, due to their Er³+ intra-4f emission at 1.54 μm, a standard telecommunication wavelength. Lithium niobate crystals (LN) are of great importance for fabrication of integrated optical devices, due to their excellent properties such as high electro-optical coefficient, low propagation loss, and high Curie temperature. Lithium niobate crystal has been implanted with 500 keV Er ions at a fluence of 3.0×10 15 ions/㎝² with the aim of optically doping the material in the near surface region. In order to recrystallize the amorphized implanted region the sample was annealed at 500℃ for 90 min in oxygen atmosphere. Photoluminescence (PL) and Rutherford backscattering spectrometry studies were performed on the as-implanted sample before and after annealing. 1.54 μm room-temperature photoluminescence emission was observed in the annealed sample. The relationship between annealing temperature and photoluminescence intensity is discussed.

A New Chromone Glycoside from Rhododendron spinuliferum

Gang Chen,Xue Feng Li,Qi Zhang,Hui Zi Jin,Yun Heng Shen,Shi Kai Yan,Wei Dong Zhang 대한약학회 2008 Archives of Pharmacal Research Vol.31 No.8

A new chromone glycoside, 3,5,7-trihydroxylchromone-3-O-α-L-arabinopyranoside (1), together with quercetin (2), (+)-catechin (3), (-)-epi-catechin (4) were isolated from the aerial parts of Rhododendron spinuliferum. The structure of 1 was elucidated on the basis of spectroscopic and 2D-NMR spectral analysis. In addition, 1 exhibited mild inhibitory effect on NO production in LPS-stimulated RAW264.7 cells.