R. Hernandez-Ortiz, L. P. Montejano, J. A. Rodriguez-Velazquez
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.
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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…
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.
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。
Italian domination in generalized Petersen graphs
在图$G=(V,E)$中，每个顶点$v\in V$用$0$、$1$或$2$标记，使得每个用$0$标记的顶点与至少一个标记为$2$的顶点或两个标记为$1$的顶点相邻。这种标记被称为$G$的意大利支配函数（IDF）。IDF $f$的重量为$w(f)=\sum_{v\in V}f(v)$。$G$的意大利控制数是$γ_{I}(G)=\min_{f} w(f)$。Gao等人（2019）已确定了$γ_I(P(n,3))$的价值。本文主要研究了广义Petersen图$P(n, k)$，$k\neq3$的意大利控制数.我们确定了$k\ge4$、$k\equiv2,3(\bmod5)$和$n\equiv0(\bmod5)$的$γ_I(P(n, 1))$、$γ_I(P(n, 2))$和$γ_I(P(n, k))$的值。对于其它$P(n,k)$，我们给出了$γ_I(P(n, k))$的一个界.利用所得到的结果，我们通过给出$P(n,1)$是$γ_I=γ_{r2}$的一个例子并表征$P(n,2)$，其中$γ_I(P(n,2))=γ_{r2}(P(n,2))$部分地解决了Brešar等人（2007）提出的公开问题。此外，我们的结果意味着$P(n,1)$ $(n\equiv0(\bmod\ 4))$是意大利语，$P(n,1)$ $(n\not\equiv0(\bmod\ 4))$和$P(n,2)$不是意大利语…
Domination related parameters in rooted product graphs
一个图$G$的顶点集$S$是$G$中的控制集，如果$S$之外的每个顶点都与至少一个属于$S$的顶点相邻。$G$的控制参数与满足某些控制性质的图的顶点集以及$G$的顶点上的其它条件有关。在这里，我们研究了几个控制相关的参数在根积图。
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.
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…
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…
Covering Italian domination in graphs
对于图$G=(V(G),E(G))$，$G$的意大利控制函数（ID函数）是函数$f:V(G)\rightarrow \{0,1,2\}$，使得对于每个顶点$v\in V(G)$具有$f(v)=0$，$f(N(v))\geq2$，即，存在顶点$u \in N(v)$具有$f (u) = 2$，或者存在两个顶点$x,y\in N(v)$具有$f(x)=f(y)=1$。函数$f:V(G)\rightarrow \{0,1,2\}$是$G$的覆盖意大利控制函数（CID函数），如果$f$是ID函数，并且$\{v\in V(G)\mid f(v)\neq0\}$是顶点覆盖集。覆盖意大利控制数（CID数）$γ_{cI}(G)$是在$G$的所有CID函数上采取的最小权重。本文研究了图的CID数。我们表明，这个参数的计算问题是NP-困难的，即使限制到一些著名的家庭的图，并找到一些界限这个参数。我们刻画了图的CID数达到其顶点覆盖数的上界两倍的图族以及CID数达到其阶数的下界一半的所有无爪图。我们也给出了一些具有小或大CID数的图族的特征。
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。
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。
Independent Double Roman Domination on Block Graphs
给定一个图$G=(V,E)$，如果当$f(v)=0$时$f$满足$\sum_{u \in N(v)}f(u) \geq 2$，则$f:V \rightarrow \{0,1,2 \}$是$G$的意大利控制函数.将$w(f)=\sum_{v \in V}f(v)$表示为$f$的重量。设$V_i=\{v:f(v)=i\},i=0,1,2$，如果$V_1 \cup V_2$是独立集，则称$f$为独立意大利控制函数. $G$的独立意大利控制数是独立意大利控制函数$f$的最小权，记为$i_{I}(G)$。将连通块图$G$的独立控制问题等价地转化为其块割点图$T$的诱导独立控制问题，并给出了求解任意连通块图$G$的$i_{I}(G)$的基于动态规划的线性时间算法.
Perfect Italian domination on planar and regular graphs
图$G=(V,E)$的完美意大利控制函数是函数$f : V \to \{0,1,2\}$，使得对于每个顶点$f(v) = 0$，它保持$\sum_{u \in N(v)} f(u) = 2$，即，由$f$分配给$v$的邻居的标签的权重正好是2。一个完美意大利函数的权是顶点权的和。$G$的完美意大利控制数，记作$γ^p_I(G)$，是$G$的任何完美意大利控制函数的最小权。在引入参数的同时，Haynes和Henning（离散应用数学（2019），164- 177）还提出了确定最佳可能常数$c_\mathcal{G}$的问题，使得当$G$在特定的图类$\mathcal{G}$中时，对于所有$n$阶的图，$γ^p_I(G) \leq c_\mathcal{G} \times n$。他们证明了当$\mathcal{G}$是二部图类时，$c_\mathcal{G} = 1$是二部图，并提出了平面图和正则图的问题。我们解决了他们的问题，正是平面图证明$c_\mathcal{G} = 1$和立方图证明$c_\mathcal{G} = 2/3$。对于分裂图，我们还表明，$c_\mathcal{G} = 1$。此外，我们还刻画了$γ^p_I(G)$等于2和3的图$G$，并确定了几个简单结构图的参数的精确值。我们的结论是证明它是NP-完全的决定是否一个给定的二部平面图承认一个完美的意大利控制功能的重量…
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。
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。
Balanced Domination in Convex Polytopes, Trees, and Grid Graphs
该文解决图的平衡控制数相关公开问题，证明凸多面体Aₙ、Dₙ、Rn''等为d-平衡图，给出两层后代根树的d-平衡特征，证明满二叉树为d-平衡图，确定网格图的精确平衡控制数。
Cyclic Sources of Strong Domination in Graph Norms
针对Conlon和Lee提出的问题，构造了双参数族非半范数2-连通强支配图，通过双根块循环融合实现，确定K_{2,m}的循环融合准则，还分类了相关外平面图强支配图及块。
Paired domination in trees: A linear algorithm and asymptotic normality
本文提出计算树的配对支配数的线性算法，并证明随机有根树（由条件Galton-Watson过程生成）及随机Cayley树的配对支配数具有渐近正态性，其中Cayley树的期望约为0.5177n。
An \textit{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 $f(w) = 2$. The \textit{weight} of an Italian dominating function $f$ is the value $ω(f) = f(V(D)) = \sum_{u \in V(D)} f(u)$. The \textit{Italian domination number} of a digraph $D$, denoted by $γ_I(D)$, is the minimum taken over the weights of all Italian dominating functions on $D$. The \textit{Italian bondage number} of a …
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。
It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove real-rootedness.` Recently I. Beaton and J. Brown (2020) proved the for almost all graphs the coefficients of the domination polynomial form a unimodal sequence, and C. Barton, J. Brown and D. Pike (2020) proved that the forest polynomial (aka acyclic polynomial) is real-rooted iff $G$ is a forest. Let $\mathcal{A}$ be a graph property, and let $a_i(G)$ be the num…
For positive integers $n$ and $k$, the dendrimer $T_{n, k}$ is defined as the rooted tree of radius $n$ whose all vertices at distance less than $n$ from the root have degree $k$. The dendrimers are higly branched organic macromolecules having repeated iterations of branched units that surroundes the central core. Dendrimers are used in a variety of fields including chemistry, nanotechnology, biology. In this paper, for any positive integer $\ell$, we count the number of paths of length $\ell$ of $T_{n, k}$. As a consequence of our main results, we obtain the average distance of $T_{n, k}$ whi…
Further results on outer independent $2$-rainbow dominating functions of graphs
本文证明连通无爪图的OI$2$RD数下界为$n/2$并刻画等号成立图类，解决开放问题；同时给出有根积与冠积的精确公式及笛卡尔积、直接积的上界。
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