Anu V., Aparna Lakshmanan S.
In this paper, we study the double Roman domination number of generalized Sierpiński graphs $S(G,t)$. More precisely, we obtain a bound for the double Roman domination number of $S(G, t)$. We also find the exact value of $γ_{dR}(S(K_{n}, 2))$.
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Different types of domination on the Sierpiński graphs S(K_n,t) will be studied in this paper. More precisely, we propose a minimal dominating set for S(K_n,t) so that the exact values of their domination numbers, Roman domination numbers, and double Roman domination numbers are given. As applications, some previous bounds and results are confirmed to be tight and further generalized.
A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G=(V,E)$ if for every vertex $v\in V$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$. In this article we study the Roman domination number of Generalized Sierpiński graphs $S(G,t)$. More precisely, we obtain a general upper bound on the Roman domination number of $S(G,t)$ and we discuss the tightnes…
We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating function. Our characterization is based on five simple extension operations, and reveals several structural properties of these trees.
In this paper we determine the signed Roman domination number of the join of cycles, wheels, fans and friendship graphs.
Bounds for (strong) Roman $k$-dominations
本文研究了连通图的罗马k-支配和强罗马k-支配数的精确上界，统一并扩展了k=2和k=3时的已有结果，并精确刻画了达到这些上界的图。
The Sierpiński Domination Number
本文定义图的 Sierpiński 积的下、上 Sierpiński 支配数，建立一般界，并针对两个环的乘积给出上支配数及部分下支配数的精确结果。
In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination number, and the total restrained domination number is equal to the total domination number. A number of open problems are posed.
A Roman domination function on a graph G is a function $r:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $r(u)=0$ is adjacent to at least one vertex $v$ for which $r(v)=2$. The weight of a Roman function is the value $r(V(G))=\sum_{u\in V(G)}r(u)$. The Roman domination number $γ_R(G)$ of $G$ is the minimum weight of a Roman domination function on $G$. "Roman Criticality" has been defined in general as the study of graphs where the Roman domination number decreases when removing an edge or a vertex of the graph. In this paper we give further results in this topic as…
A \emph{perfect Roman dominating function} (PRDF) on a graph $G = (V, E)$ is a function $f : V \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The weight of a PRDF is the value $w(f) = \sum_{u \in V}f(u)$. The minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman domination number $γ_R^p(G)$} of $G$. A graph $G$ is perfect Roman domination domination stable if the perfect Roman domination number of $G$ remains unchanged under the removal of any vertex. In this paper, we …
Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that $f(u)=2$. The Roman domination number of $G$ is the minimum weight $ω(f)=\sum_{x\in V(G)}f(x)$ among all Roman dominating functions $f$ on $G$. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving…
Domination game and total domination game played on Sierpiński graphs
该研究聚焦谢尔宾斯基图上的控制与全控制博弈，通过推导严格不等式改进了控制博弈数的平凡界，并确立了全控制博弈数的下界，丰富了图论博弈领域的相关结论。
For a graph $G$, let $γ_{r2}(G)$ and $γ_R(G)$ denote the $2$-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) proved $γ_{r2}(G)+γ_R(G)\leq \frac{6}{4}n(G)$ for a connected graph $G$ of order $n(G)$ at least $3$. Furthermore, they conjectured $γ_{r2}(G)+γ_R(G)\leq \frac{4}{3}n(G)$ for a connected graph $G$ of minimum degree at least $2$ that is distinct from $C_5$. We characterize all extremal graphs for their inequality and prove th…
The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which $f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on $G$. In this…
A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The $weight$ of a Roman domination function $f$ is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the $Roman\ domination\ number$ of $G$, denoted by $γ_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $γ_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n …
Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27-32) we prove $γ_{r2}(G)\leq 2γ_r(G)$ for every graph $G$, where $γ_{r2}(G)$ and $γ_r(G)$ denote the $2$-rainbow domination number and the weak Roman domination number of $G$, respectively. We characterize the extremal graphs for this inequality that are $\{ K_4,K_4-e\}$-free, and show that the recognition of the $K_5$-free extremal graphs is NP-hard.
On a generalization of Roman domination with more legions
本文将 Roman 与 Italian 控制推广为任意正整数 k 下的强 Roman 及 Roman k-控制，比较其控制数，并首次确定完全二部图上的相关控制数。
An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ γ_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f…
A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. The minimum cardinality of a dominating set for $G$ is called the domination number of $G$. A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G$ if for every vertex $v$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination…
For any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least $k$ and for any vertex $v$ with $f(v)\neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$. The minimum weight of a Roman $k$-tuple dominating function $f$ on $G$ is called the Roman $k$-tuple domination number of the graph where the weight of $f$ is $f(V)=\sum_{v\in V}f(v)$. In this paper, we initiate to study the Roman $k$-tuple domination numbe…
For a graph G=(V,E), a restrained double Roman dominating function is a function f:V\rightarrow\{0,1,2,3\} having the property that if f(v)=0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then the vertex v must have at least one neighbor w with f(w)\geq2, and at the same time, the subgraph G[V_0] which includes vertices with zero labels has no isolated vertex. The weight of a restrained double Roman dominating function f is the sum f(V)=\sum_{v\in V}f(v), and the minimum weight of a restrained double Roman dominating functi…
Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman domination in graphs was described. 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 this paper we initiate the…
For a graph $G=(V,E)$ of order $n$, a Roman $\{2\}$-dominating function $f:V\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$, or $v$ is adjacent to least two vertices assigned $1$ under $f$. In this paper, we classify all graphs with Roman $\{2\}$-domination number belonging to the set $\{2,3,4,n-2,n-1,n\}$. Furthermore, we obtain some results about Roman $\{2\}$-domination number of some graph operations.
Several Roman domination graph invariants on Kneser graphs
本文研究了Kneser图上的罗马支配、全罗马支配和符号罗马支配三个图不变量，给出了相应支配数的精确值及上下界。
Triple Roman Domination in Graphs
本文研究图中的三重罗马支配，要求每城至少三军团防御，给出一般图的界、部分图族的精确值及复杂性结果。
A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function $f : V \rightarrow \{0,1,2\}$ satisfying the following: - every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) =2$, and - if $x$ is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then $f(x)=1$. The weight of a quasi-total Roman dominating function is the value $ω(f)=f(V)=\sum_{u\in V} f(u)$. The minimum weight of a quasi-total Roman dominating function on a graph $G$ is called the quasi-total Roman domination number of $G$. We introduce th…
Roman domination number of zero-divisor graphs over commutative rings
本文研究交换环上零因子图$\Gamma(R)$的Roman控制数，推导了$T(\Gamma(R))$的Roman控制数的界，丰富了零因子图的控制理论相关结果。
Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$
本文研究了柱形图C_m□P_n的[k]-罗马 domination 数，通过线性周期构造、均匀天花板型标记和基于打包的改进三种方法，推导出新的上界，并比较了不同参数k下的效率。
For a given graph $G$ without isolated vertex we consider a function $f: V(G) \rightarrow \{0,1,2\}$. For every $i\in \{0,1,2\}$, let $V_i=\{v\in V(G):\; f(v)=i\}$. The function $f$ is known to be an outer-independent total Roman dominating function for the graph $G$ if it is satisfied that; (i) every vertex in $V_0$ is adjacent to at least one vertex in $V_2$; (ii) $V_0$ is an independent set; and (iii) the subgraph induced by $V_1\cup V_2$ has no isolated vertex. The minimum possible weight $ω(f)=\sum_{v\in V(G)}f(v)$ among all outer-independent total Roman dominating functions for $G$ is ca…
For a graph $G$, let $γ_R(G)$ and $γ_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $γ_{r2}(G)\leq γ_R(G)\leq \frac{3}{2}γ_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $γ_R(G)-γ_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs efficiently. We show that for every fixed…
Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs
本文研究字典序乘积图的完美控制数、罗马控制数和完美罗马控制数，给出前两者的闭式公式，为完美罗马控制数提供界限及达到条件，并刻画完美罗马图。
A Roman dominating function (RD-function) on a graph $G = (V(G), E(G))$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight $f(V(G))$ of a RD-function $f$ on $G$ is the value $Σ_{v\in V(G)} f (v)$. The {\em Roman domination number} $γ_{R}(G)$ of $G$ is the minimum weight of a RD-function on $G$. The six classes of graphs resulting from the changing or unchanging of the Roman domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. We consider relationships among the classes…
Theoretical results for Perfect Location signed Roman domination problem
本文研究图论中的完美定位符号罗马控制（PLSRD）问题，给出多种经典图类的PLSRD数精确值，并推导3-正则图下界与花斯诺克图上界，揭示约束与图结构的关联。
Further Results on the Majority Roman Domination in graphs
本文研究了图中的多数罗马支配问题，探讨了其性质并给出了某些图的多数罗马支配数的上下界。
An independent double Roman dominating function (IDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with assigned $3$ under $f$, and if $f(v)=1$, then there exists $w\in N(v)$ with $f(w)\geq2$ such that the positive weight vertices are independent. The weight of an IDRDF is the value $\sum_{u\in V}f(u)$. The independent double Roman domination number $i_{dR}(G)$ of a graph $G$ is the minimum weight of an IDRDF on G. We initiate the study of the …
Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $ v_{1},v_{2}\in N(v)$ such that $f(v_{1})=f(v_{2})=2$ or there exists $ w \in N(v)$ such that $f(w)=3$, and if $f(v)=1$, then there exists $ w \in N(v)$ such that $f(w)\geq 2$ is called a double Roman dominating function (DRDF). The weight of a DRDF $f$ is the sum $f(V)=\sum_{v\in V}f(v)$. The double Roman domination number, $γ_{dR}(G)$, is the minimum among the weights of DRDFs on $G$. In this paper, we study the impact of some graph operations, such as cartesian product, a…
For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is the sum $f (V) = \cup_{v\in V} f (v)$, and the minimum weight of a Roman dominating function on $G$ is the Roman domination number $γ_R(G)$ of $G$. The Roman bondage number $b_R(G)$ of $G$ is the minimum cardinality of all sets $F \subseteq E$ for which $γ_R(G - F) > γ_R(G)$. A graph $G$ is in the class $\mathcal{R}_{UVR}$ if the Roman domination number remains un…
For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of an RDF of $G$ is its Roman domination number and denoted by $γ_ R(G)$. In this paper, we first show that $γ_{R}(G)+1\leq γ_{R}(μ(G))\leq γ_{R}(G)+2$, where $μ(G)$ is the Mycielekian graph of $G$, and then characterize the graphs achieving equality in these bounds. Then for any positive integer $m$,…
On $k$-limited domination: complexity and Sierpiński graphs
本文证明了 $k$-限制支配集问题的复杂性（包括平面图情形），并确定了 Sierpiński 图的 $k$-限制支配数的精确值。
Let $G=(V,E)$ be a graph of order $n$ and let $γ_{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $δ\geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $γ_{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k…
Integer Linear Programming Formulations for Triple and Quadruple Roman Domination Problems
针对NP-hard的三重、四重罗马支配问题，本文提出对应的遗传算法与整数线性规划公式，通过IBM CPLEX 22.1求解器在NetworkX Erdos-Renyi随机图上完成了模型实现与结果获取。
Roman $\{2\}$-domination on Graphs with "few" 4-paths
本文通过模分解分析，研究了在具有有限数量4-路径的图类上的罗马{2}-控制问题，给出了蜘蛛图、良好标记蜘蛛图及某些关键素分裂图的罗马{2}-控制数，并提供了计算余图、P4-稀疏图、P4-整洁图和伙伴限制图的罗马{2}-控制数的线性时间算法，同时证明了罗马{2}-控制在P4-负载图上的NP完全性。
In this paper we deal with the calculation of the signed (total) Roman domination numbers, $γ_{sR}$ and $γ_{stR}$ respectively, on a few classes of planar graphs from the literature. We give proofs for the exact values of the numbers $γ_{sR}(A_n)$ and $γ_{sR}(R_n)$ as well as the numbers $γ_{stR}(S_n)$ and $γ_{stR}(T_n)$. For some other classes of planar graphs, such as $Q_n$, %$S_n"$ and $T_n"$, lower and upper bounds on $γ_{sR}$ are calculated and proved. %We give some open problems on the exact values of $γ_{sR}$ and $γ_{stR}$ for some classes of planar graphs.
An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of $G$, denoted by $γ'_R(G)$, is the minimum weight $w(f) = \sum_{e\in E(G)} f(e)$ of an edge Roman dominating function $f$ of $G$. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if $G$ is a graph of maximum degree $Δ$ on $n$ vertices, then $γ_R'(G) \le \lceil \fracΔ{Δ+1} n \rceil$. While t…
A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Σ_{v\in V} f(v)$. The Roman domination number, $γ_R(G)$, of $G$ is the minimum weight of an RDF on $G$. An RDF of minimum weight is called a $γ_R$-function. A graph G is said to be $γ_R$-excellent if for each vertex $x \in V$ there is a $γ_R$-function $h_x$ on $G$ with $h_x(x) \not = 0$. We present a constructive characterization of $γ_R$-excellent trees using labelings. A graph $G$ is…
A dual view of Roman Domination: The 2-limited packing problem
本文将罗马 domination 问题与 2 限制打包问题进行比较，证明二者在树结构上为强对偶，从而通过解决 2 限制打包问题来解决罗马 domination 问题。
Let $G=(V(G),E(G))$ be a simple graph. A restrained double Roman dominating function (RDRD-function) of $G$ is a function $f: V(G) \rightarrow \{0,1,2,3\}$ satisfying the following properties: if $f(v)=0$, then the vertex $v$ has at least two neighbours assigned 2 under $f$ or one neighbour $u$ with $f(u)=3$; and if $f(v)=1$, then the vertex $v$ must have one neighbor $u$ with $f(u) \geq 2$; the induced graph by vertices assigned 0 under $f$ contains no isolated vertex. The weight of a RDRD-function $f$ is the sum $f(V)=\sum_{v \in V(G)} f(v)$, and the minimum weight of a RDRD-function on $G$ …
Signed Total Roman Domination and Domatic Numbers: Degree Three and Complete Multipartite Graphs
研究三次图的符号全罗马支配数与其开放包装数、2-元全支配数和符号全支配数的关系，得到紧界和NP完全性结果，并确定完全多部图的精确值。
Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is a total Roman $\{2\}$-dominating function if: (1) every vertex $v\in V$ for which $f(v)=0$ satisfies that $\sum_{u\in N(v)}f(u)\geq 2$, where $N(v)$ represents the open neighborhood of $v$, and (2) every vertex $x\in V$ for which $f(x)\geq 1$ is adjacent to at least one vertex $y\in V$ such that $f(y)\geq 1$. The weight of the function $f$ is defined as $ω(f)=\sum_{v\in V}f(v)$. The total Roman $\{2\}$-domination number, denoted by $γ_{t\{R2\}}(G)$, is the minimum weight among all total Roman $\{2\}$-dominating functions on $G$.…
A total Roman dominating function on a graph $G$ is a function $% f:V(G)\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to some vertex $u$ with $f(u)=2$, and the subgraph of $G$ induced by the set of all vertices $w$ such that $f(w)>0$ has no isolated vertices. The weight of $f$ is $Σ_{v\in V(G)}f(v)$. The total Roman domination number $γ_{tR}(G)$ is the minimum weight of a total Roman dominating function on $G$. A graph $G$ is $k$-$γ_{tR}$-edge-critical if $γ_{tR}(G+e)<γ_{tR}(G)=k$ for every edge $e\in E(\overline{G})\neq \emptyset $, and $k$-$γ_{tR}$-edge-supercri…
In this paper we obtain closed formulae for several parameters of generalized Sierpiński graphs $S(G,t)$ in terms of parameters of the base graph $G$. In particular, we focus on the chromatic, vertex cover, clique and domination numbers.
As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the n…
$[k]$-Roman domination on cylindrical grids $C_m \Box P_n$
本文研究了圆柱形网格 $C_m \Box P_n$ 上的 $[k]$-罗马 domination 数，推导了上界并揭示了不同构造在不同参数下的主导范围。
A sequence $S=(v_1,\ldots,v_k)$ of distinct vertices of a graph $G$ is called a legal sequence if $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j]\not=\emptyset$ for any $i$. The maximum length of a legal (dominating) sequence in $G$ is called the Grundy domination number $γ_{gr}(G)$ of a graph $G$. It is known that the problem of determining the Grundy domination number is NP-complete in general, while efficient algorithm exist for trees and some other classes of graphs. In this paper we find an efficient algorithm for the Grundy domination number of an interval graph. We also show the exact value of…
A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a function $f: V \longrightarrow \{0,1,2\}$ if $f(v)=0$ and $f(u)=0$ for every vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating function if for every $v$ such that $f(v)=0$ there exists a vertex $u$ adjacent to $v$ such that $f(u)\in \{1,2\}$ and the function $f': V \longrightarrow \{0,1,2\}$ defined by $f'(v)=1$, $f'(u)=f(u)-1$ and $f'(z)=f(z)$ for every $z\in V \setminus\{u,v\}$, has no undefended vertices. The weight of $f$ is $w(f)=\sum_{v\in V(G) }f(v)$. The weak Roman domination number …
Roman-Type Domination on Convex and Chordal Bipartite Graphs: Algorithms and Hardness
研究凸二分图和弦二分图上四种罗马支配变体，为凸二分图开发统一动态规划框架，可\(O(n^6)\)时间计算相关参数，同时证明弦二分图上部分变体仍NP完全，明确两类图算法差异。
Theoretical Studies of the k-Strong Roman Domination Problem
本文首次对k-强罗马控制问题进行理论研究，提出最小权重映射方法，为多类图给出精确值、界及三次图下界。
p-Strong Roman Domination in Graphs
本研究提出图的p-强罗马控制新参数，定义对应的函数与控制数，证明该问题的NP完全性，给出依赖经典图不变量的紧致上下界，并确定几类图的精确值。
Double Italian domination in trees
本文研究树上双重意大利 domination 数，发现其与双重罗马 domination 数不同，并给出了非平凡树的精确界及达到该界的树特征。
Further results on \([k]\)-Roman domination on cylindrical grids \(C_m \Box P_n\)
本文研究圆柱形图$C_m \Box P_n$的$[k]$-罗马 domination 数，通过局部邻域约束得出下界，并利用与高效 domination 的联系刻画最优配置，构造周期性函数得到上界，经优化后展示其依赖于参数$k$和路径长度的强度。
Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles
本文研究路径与圈的笛卡尔积的外独立罗马控制数，确定了部分参数下的精确值，并给出了n≥4、m≥4时的上界，丰富了图的罗马控制理论相关结果。
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