Christopher M. van Bommel
An Italian dominating function on a (di)graph $G$ with vertex set $V(G)$ is a function $f: V(G) \to \{0, 1, 2\}$ such that every vertex $v \in V(G)$ such that $f(v) = 0$ has an (in)neighbour assigned 2 or two (in)neighbours assigned 1. We complete the investigation of the Italian domination numbers of Cartesian products of directed cycles.
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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…
In this article, we obtain closed formulae for the Italian domination number of rooted product graphs. As a particular case of the study, we derive the corresponding formulas for corona graphs, and we provide an alternative proof that the problem of computing the Italian domination number of a graph is NP-hard.
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(G) $ 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(G)}f(v) $ and Italian domination number, $ γ_I(G) $ is the minimum weight of an IDF. In this paper, we study the impact of corona operator and addition of twins on Italian domination number.
In this paper, an upper bound for the perfect Italian domination number of the cartesian product of any two graphs is obtained and the exact value of this parameter for cartesian product of some special graphs are obtained. We have also proved that for any two positive integers $a$, $b$ there exists a graph $G$ and an induced subgraph $H$ of $G$ such that $γ_I^p(G) = a$ and $γ_I^p(H) = b$. Relationship of the perfect Italian domination number with the Roman domination number and the perfect domination number of a graph $G$ are obtained and the corresponding realization problems are also solved…
A Graph Theoretical Approach to Optimizing Minimum Italian Domination Sets
本文针对Toeplitz图族，分析意大利支配数的变化规律，为基于该问题的网络安全策略提供图论层面的支撑。
In this paper, we show that the Italian domination number of every lexicographic product graph $G\circ H$ can be expressed in terms of five different domination parameters of $G$. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let $N(v)$ denote the open neighbourhood of $v\in V(G)$, and let $w=(w_0,w_1, \dots,w_l)$ be a vector of nonnegative integers such that $ w_0\ge 1$. We say that a function $f: V(G)\longrightarrow \{0,1,\dots ,l\}$ is a $w$-dominating function if $f…
Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textit{restrained domination number}, denoted by $γ_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. A function $f : V \rightarrow \{0, 1, 2\}$ is a \textit{restrained Italian dominating function} on $G$ if (i) for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N_G(v)} f(u) \geq 2$, (ii) the subgraph induced by $\{v \in V \mid f(v)=0 \}$ has no isolated vert…
Given a graph $G$ and a subset of vertices $D\subseteq V(G)$, the external neighbourhood of $D$ is defined as $N_e(D)=\{u\in V(G)\setminus D:\, N(u)\cap D\ne \varnothing\}$, where $N(u)$ denotes the open neighbourhood of $u$. Now, given a subset $D\subseteq V(G)$ and a vertex $v\in D$, the external private neighbourhood of $v$ with respect to $D$ is defined to be $epn(v,D)=\{u\in V(G)\setminus D: N(u)\cap D=\{v\}\}.$ The strong differential of a set $D\subseteq V(G)$ is defined as $\partial_s(D)=|N_e(D)|-|D_w|,$ where $D_w=\{v\in D: epn(v,D)\neq \varnothing\}$. In this paper we focus on the st…
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。
In a graph $G=(V,E)$, each vertex $v\in V$ is labelled with $0$, $1$ or $2$ such that each vertex labelled with $0$ is adjacent to at least one vertex labelled $2$ or two vertices labelled $1$. Such kind of labelling is called an Italian dominating function (IDF) of $G$. The weight of an IDF $f$ is $w(f)=\sum_{v\in V}f(v)$. The Italian domination number of $G$ is $γ_{I}(G)=\min_{f} w(f)$. Gao et al. (2019) have determined the value of $γ_I(P(n,3))$. In this article, we focus on the study of the Italian domination number of generalized Petersen graphs $P(n, k)$, $k\neq3$. We determine the value…
An Italian dominating function on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. In this paper, we determine the exact values of the Italian domination numbers of some products of directed cycles.
In this paper, we prove a problem proposed by Brešar: for any graphs $G$ and $H$, $Γ(G\square H)\geΓ(G)Γ(H)+ \min\{|V(G)|-Γ(G),|V(H)|-Γ(H)\}$, where $Γ(G)$ denotes the upper domination number of $G$.
For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ 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 $f$. The weight of an ID function is $\sum_{v\in V(G)}f(v)$. The Italian domination number is the minimum weight taken over all ID functions of $G$. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) $f$ of $G$ is an ID function of $G$ for w…
The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Mobius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined, the proof has either been trivial, or has been derived from lower bounds established by considering different forms of domination. However, the latter mode of proof is not applicable for most graphs, including those considered here. Hence, this work represents the first attempt to determine secure domination numbers via the properties of secure domination itse…
For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) of $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ such that for each vertex $v\in V(G)$ with $f(v)=0$, $f(N(v))\geq2$, that is, either there is a vertex $u \in N(v)$ with $f (u) = 2$ or there are two vertices $x,y\in N(v)$ with $f(x)=f(y)=1$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a covering Italian dominating function (CID function) of $G$ if $f$ is an ID function and $\{v\in V(G)\mid f(v)\neq0\}$ is a vertex cover set. The covering Italian domination number (CID number) $γ_{cI}(G)$ is the minimum weight taken ove…
In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of Nordhaus-Gaddum type involving secure domination and weak Roman domination. Furthermore, the particular case of Cartesian product graphs is considered.
Ho proved in [A note on the total domination number, Util.Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs with no isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and $K_2$ or $C_n$, Util. Math. 83 (2010) 313--322] by characterizing the pairs of graphs $G$ and $H$ for which $γ_t(G\Box H)=\frac{1}{2}γ_t(G) γ_t(H)\,$, whenever $γ_t(H)=2$. In addition, we present an infinite family of graphs $G_n$ with $γ_t(G_n)=2n…
On the total Italian domination number in digraphs
本文研究有向图的全意大利支配数，给出其上下界，并确定笛卡尔积 $P_2\Box P_n$ 和 $P_3\Box P_n$ 的精确全意大利支配数。
Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$ using this process is known as a power dominating set of $G$, and the power domination number of $G$, $γ_P(G)$, is the minimum number of vertices in a power dominating set. We introduce a new partition on the vertices of a graph to provide a lower bound for the power domination number. We also consider the power domination number of the Cartesian product of …
Let $γ(C_m\Box C_n)$ be the domination number of the Cartesian product of directed cycles $C_m$ and $C_n$ for $m,n\geq2$. Shaheen [] and Liu and al.[ ], [ ] determined the value of $γ(C_m\Box C_n)$ when $m \leq 6$ and when both $m$ and $n$ $\equiv 0$ $(mod\: 3)$. In this article we give, in general, the value of $γ(C_m\Box C_n)$ when $m\equiv 2$ $(mod\: 3)$ and improve the known lower bound for most of the remaining cases. We also disprove the conjectured formula for the case $m$ $\equiv 0$ $(mod\: 3)$ appearing in \cite{}
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。
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…
Let $G=(V,E)$ be a finite undirected graph. A set $S$ of vertices in $V$ is said to be total $k$-dominating if every vertex in $V$ is adjacent to at least $k$ vertices in $S$. The total $k$-domination number, $γ_{kt}(G)$, is the minimum cardinality of a total $k$-dominating set in $G$. In this work we study the total $k$-domination number of Cartesian product of two complete graphs which is a lower bound of the total $k$-domination number of Cartesian product of two graphs. We obtain new lower and upper bounds for the total $k$-domination number of Cartesian product of two complete graphs. Som…
The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $γ_P(G)$, is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In \cite{ZKC_cubic}, Zhao et al. proved that if $G$ is a connected claw-free cubic graph of order $n$, then $γ_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $γ_P(G) \le n/6$, a…
Given a graph $G$, a dominating set $D$ is a set of vertices such that any vertex in $G$ has at least one neighbor (or possibly itself) in $D$. A ${k}$-dominating multiset $D_k$ is a multiset of vertices such that any vertex in $G$ has at least $k$ vertices from its closed neighborhood in $D_k$ when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) and properties of binary matrices to prove a "Vizing-like" inequality on minimum ${k}$-dominating multisets of graphs $G,H$ and the Cartesian product graph $G \Box H$. Specifically, denoting the siz…
In this paper, we consider the connected power domination number ($γ_{P, c}$) of three standard graph products. The exact value for $γ_{P, c}(G\circ H)$ is obtained for any two non-trivial graphs $G$ and $H.$ Further, tight upper bounds are proved for the connected power domination number of the Cartesian product of two graphs $G$ and $H.$ Consequently, the exact value of the connected power domination number of the Cartesian product of some standard graphs is determined. Finally, the connected power domination number of tensor product of graphs is discussed.
On $k$-limited domination: complexity and Cartesian products
本文证明对于每个固定整数 $k\ge 2$，判定图是否存在大小不超过 $\ell$ 的 $k$-限制定支配集是 NP-完全的，并系统研究了笛卡尔积中的 $k$-限制定支配，给出了上下界及精确值。
A general lower bound for the domination number of cylindrical graphs
本文提出了路与圈的笛卡尔积控制数的一个下界，在圈长为5的倍数时紧致，并改进了此前已知的最佳自然下界。
A digraph $D$ is an efficient closed domination digraph if there exists a subset $S$ of $V(D)$ for which the closed out-neighborhoods centered in vertices of $S$ form a partition of $V(D)$. In this work we deal with efficient closed domination digraphs among several product of digraphs. We completely describe the efficient closed domination digraphs among lexicographic and strong products of digraphs. We characterize those direct products of digraphs that are efficient closed domination digraphs, where factors are either two cycles or two paths. Among Cartesian product of digraphs, we describe…
The 2-rainbow domination number of Cartesian product of cycles
本文研究了两个圈的笛卡尔积的2-彩虹控制数，给出了若干无穷族的精确值，并证明了所有其他情况的上界与下界。
Given a graph $G=(V,E)$, $f:V \rightarrow \{0,1,2 \}$ is the Italian dominating function of $G$ if $f$ satisfies $\sum_{u \in N(v)}f(u) \geq 2$ when $f(v)=0$. Denote $w(f)=\sum_{v \in V}f(v)$ as the weight of $f$. Let $V_i=\{v:f(v)=i\},i=0,1,2$, we call $f$ the independent Italian dominating function if $V_1 \cup V_2$ is an independent set. The independent Italian domination number of $G$ is the minimum weight of independent Italian dominating function $f$, denoted by $i_{I}(G)$. We equivalently transform the independent domination problem of the connected block graph $G$ to the induced indepe…
A $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$; the minimum size of a $k$TDS is denoted $γ_{\times k,t}(G)$. We give a Vizing-like inequality for Cartesian product graphs, namely $γ_{\times k,t}(G) γ_{\times k,t}(H) \leq 2k γ_{\times k,t}(G \Box H)$ provided $γ_{\times k,t}(G) \leq 2kρ(G)$, where $ρ$ is the packing number. We also give bounds on $γ_{\times k,t}(G \Box H)$ in terms of (open) packing numbers, and consider the extremal case of $γ_{\times k,t}(K_n \Box K_m)$, i.e., the rook'…
A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to \{0,1,2\}$ such that for every vertex $f(v) = 0$, it holds that $\sum_{u \in N(v)} f(u) = 2$, i.e., the weight of the labels assigned by $f$ to the neighbors of $v$ is exactly two. The weight of a perfect Italian function is the sum of the weights of the vertices. The perfect Italian domination number of $G$, denoted by $γ^p_I(G)$, is the minimum weight of any perfect Italian dominating function of $G$. While introducing the parameter, Haynes and Henning (Discrete Appl. Math. (2019), 164--177) also proposed the…
The orientable domination number, ${\rm DOM}(G)$, of a graph $G$ is the largest domination number over all orientations of $G$. In this paper, ${\rm DOM}$ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ${\rm DOM}(K_{n_1,n_2,n_3})$ for arbitrary positive integers $n_1,n_2$ and $n_3$. While considering the orientable domination numb…
A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $γ(G)$ is the size of a minimum dominating set in $G$. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $γ(G) γ(H) \leq γ(G \Box H)$, and Clark and Suen (2000) proved that $γ(G) γ(H) \leq 2γ(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ t…
Fuzzy incidence graphs (FIG) model real world problems efficiently when there is an extra attribute of vertex-edge relationship. The article discusses the operations on Fuzzy incidence graphs. The join, Cartesian product, tensor product, and composition of FIGs are explored. The study is concentrated mainly on strong fuzzy incidence graphs (SFIG). The idea of strong incidence domination (SID) is used, and strong incidence domination number (SIDN) in operations is examined. Basic properties of FIGs obtained from the operations are studied. Bounds for the domination number of product of two SFIG…
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。
We use a dynamic programming algorithm to establish a lower bound on the domination number of complete grid graphs of the form $C_n\square P_m$, that is, the Cartesian product of a cycle $C_n$ and a path $P_m$, for $m$ and $n$ sufficiently large.
Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$
该研究针对图上的Maker-Breaker控制博弈，推导了任意图及两顶点完全图与任意图笛卡尔积的控制数上界，精准确定了路径图P₂与Pₙ的笛卡尔积及其不交并的先手、后手Maker-Breaker控制数的精确值。
On the domination number of the cartesian product of the path graph and any pair of graphs
本文利用空间投影改进了任意一对图与路径图笛卡尔积的控制数下界，证明$γ(X\square Y\square P_2)\geq \frac{2}{3}γ(X)γ(Y)$，并推广了更大路径图$P_n$情况下的下界。
We use a dynamic programming algorithm to establish a new lower bound on the domination number of complete cylindrical grid graphs of the form $C_n\square P_m$, that is, the Cartesian product of a path and a cycle, when $n\equiv 2\pmod{5}$, and we establish a new upper bound equal to the lower bound, thus computing the exact domination number for these graphs.
Grundy Total Domination and Skew Zero Forcing in Cartesian Products of Paths and Cycles
该研究确定了路径与路径、路径与循环、循环与循环的笛卡尔积图的Grundy全支配数与斜零强制数，填补了相关界的空白并给出匹配的上下界。
Let $G$ be a graph without isolated vertices. The total domination number of $G$ is the minimum number of vertices that can dominate all vertices in $G$, and the paired domination number of $G$ is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes, i.e., the Cartesian product of two cycles $C_n$ and $C_m$ for any $n\ge 3$ and $m\in\{3,4\}$, and gives some upper bounds for $n, m\ge 5$.
2-limited broadcast domination on grid graphs
本研究针对路径与环构成的各类网格图，通过显式构造和线性规划对偶方法，分别推导其2-限制广播支配数的上下界，为该图类的广播支配问题提供了定量边界结果。
Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph
本文刻画了当 $H$ 为完全图（阶数≥3）或完全二部图时，笛卡尔积 $G\Box H$ 为高效开支配图的图 $G$，基于弱划分给出构造性刻画，并推广至 $H$ 为 4-圈或 5-圈的情形。
Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.
The power domination number arises from the monitoring of electrical networks and its determination is an important problem. Upper bounds for power domination numbers can be obtained by constructions. Lower bounds for the power domination number of several families of graphs are known, but they usually arise from specific properties of each family and the methods do not generalize. In this paper we exploit the relationship between power domination and zero forcing to obtain the first general lower bound for the power domination number. We apply this bound to obtain results for both the power d…
Counting Weight-$k$ Italian Dominating Sets on Trees and Cycles
本文研究路径、树和环上意大利支配函数的计数问题，提出动态规划算法计算所有权重系数，并给出环上系数的闭式表达式，验证了已知的意大利支配数。
A set of vertices $S$ in a simple isolate-free graph $G$ is a semi-total dominating set of $G$ if it is a dominating set of $G$ and every vertex of $S$ is within distance 2 or less with another vertex of $S$. The semi-total domination number of $G$, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semi-total dominating set of $G$. In this paper, we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs $G$ and $H$, $γ_{t2}(G\,\square\, H)\ge \frac{1}{3}γ_{t2}(G)γ_{t2}(H)$.
The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set of $G$ if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. Furt…
Maker-Breaker domination game on Cartesian products of graphs
该研究针对图笛卡尔积上的Maker-Breaker支配博弈，确定了路径、星图、完全二部图的笛卡尔积上的博弈获胜方及获胜步数，推导了星图乘积下四类博弈步数的精确公式。
Young domination on Hamming rectangles
本文研究了在汉明矩形上的年轻支配问题，提出了一种基于Young图的生长规则，并设计了近似算法，解决了L=1时的支配数计算问题。
Let $G=(V,E)$ be a simple graph. For any integer $k\geq 1$, a subset of $V$ is called a $k$-tuple total dominating set of $G$ if every vertex in $V$ has at least $k$ neighbors in the set. The minimum cardinality of a minimal $k$-tuple total dominating set of $G$ is called the $k$-tuple total domination number of $G$. In this paper, we introduce the concept of upper $k$-tuple total domination number of $G$ as the maximum cardinality of a minimal $k$-tuple total dominating set of $G$, and study the problem of finding a minimal $k$-tuple total dominating set of maximum cardinality on several clas…
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…
HPC acceleration of large (min, +) matrix products to compute domination-type parameters in graphs
本文提出基于(min, +)矩阵乘积的算法，计算路径与小阶圈图笛卡尔积的2-支配数，并通过多核CPU和GPU加速克服矩阵规模挑战，成功计算路径含12个顶点的圆柱体2-支配数。
On the $2$-domination number of cylinders with small cycles
本文研究小圈柱面图（圈与路的笛卡尔积）的 $2$-控制数，提出基于 $(\nmin,+)$ 矩阵乘积的两种算法计算 $\gamma_2(C_n\Box P_m)$（$3\leq n\leq 15$），并给出一般公式猜想。
Revisiting the outer-weakly convex domination number in graph products
本文研究了图积中外弱凸支配数的计算，分析了Cartesian、strong和lexicographic积下的图结构，并探讨了重要组合发现。
The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($γ_{\rm tcg}(G)$) $γ_{\rm cg}(G)$ of $G$. We show that $γ_{\rm tcg}(G)\in \{γ_{\rm cg}(G), γ_{\rm cg}(G) + 1, γ_{\rm cg}(G) + 2\}$, and consequently define $G$ as …
The Dual-Server Domination Number of Some Graph Operations
本文研究若干图运算的双服务器控制数，确定了联、笛卡尔积、 corona 图及路、圈、完全二部图分裂图的对应控制数，拓展了双服务器控制的研究。
Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a vertex of $U$. The domination polynomial of $G$ is the generating function whose coefficients are the number of dominating sets of a given size. We show that the domination polynomial is unimodal, i.e., the coefficients are non-decreasing and then non-increasing, for several well-known families of graphs. In particular, we prove unimodality for spider graphs with at most $400$ legs (of arbitrary length), lollipop graphs, arbitrary direct pr…
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