Jerzy Topp, Paweł Żyliński
Let $γ(G)$ and $β(G)$ denote the domination number and the covering number of a graph $G$, respectively. A connected non-trivial graph $G$ is said to be $γβ$-{perfect} if $γ(H)=β(H)$ for every non-trivial induced connected subgraph $H$ of $G$. In this note we present an elementary proof of a characterization of the $γβ$-perfect graphs.
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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 …
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…
Let $γ_g(G)$ and $γ_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $γ_g$-perfect (resp. $γ_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $γ_g(F)=γ(F)$ (resp. $γ_{tg}(F)=γ_t(F)$). A recursive characterization of $γ_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $γ_g$-perfect graphs. It is proved that every minimally $γ_g$-imperfect graph has domination number $2$. All minimally $γ_g$-imperfect triangle-free graphs are determined. It is also proved that $γ_{tg…
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…
On Camby-Plein's Characterization of Domination Perfect Graphs
该研究指出Camby和Plein提出的关于支配完美图的特征化结果存在错误，并重新证明了Zverovich和Zverovich在1995年的特征化结果。
For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $γ_p(G),$ is the minimum cardinality of a perfect dominating set of $G.$ The perfect domination number is found for knights graphs on square, rectangular, and infinite chessboards. Indeed, exact values or bounds are given for all chessboards except those with 3 rows and number of columns congruent to 1, 2, or 3 modulo 8.
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 dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The common independence number $α_c(G)$ of $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set of cardinality at least~$r$. The common independence number is squeezed between the independent domination number $i(G)$ and the independence number $α(G)$ of $G$, that is, $γ(G) \le i(G) \le α_c(G) \le α(G)$. A graph $G$ is dominat…
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。
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。
A $k-$quasiperfect dominating set ($k\ge 1$) of a graph $G$ is a vertex subset $S$ such that every vertex not in $S$ is adjacent to at least one and at most k vertices in $S$. The cardinality of a minimum k-quasiperfect dominating set in $G$ is denoted by $γ_{\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept. The quasiperfect domination chain $γ_{\stackrel{}{11}}(G)\geγ_{\stackrel{}{12}}(G)\ge\dots\geγ_{\stackrel{}{1Δ}}(G)=γ(G)$, indicates what it is lost in size when you move towards a more perfect domination…
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。
We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\R^2$. In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of $Λ$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^…
A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $γ_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge γ_ {\stackrel{}{11}}(G) \ge γ_ {\stackrel{}{12}}(G)\ge \ldots \ge γ_ {\stackrel{}{1Δ}}(G)=γ(G)$ in order…
Perfect codes in the $n$-dimensio\-nal grid $Λ_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and $v=(v_1, \ldots,v_n)$ as the graph distance $h(u,v)$ in $Λ_n$, if $|u_i-v_i|\le 1$, for all $i\in\{1, \ldots,n\}$, and as $n+1$, otherwise. Such codes are extended to multilattice graphs $Γ_n$ obtained by glueing ternary $n$-cubes along their codimension 1 ternary subcubes in such a way that each binary $n$-subcube is contained in a unique maximal lattice of $Γ_n$. T…
Perfect Edge Domination in $P_6$-free Graphs and in Graphs Without Efficient Edge Dominating Sets
本文研究了在$P_6$-free图及无高效边支配集图中的完美边支配问题，证明了相关决策问题的NP难性，并提出了一种三次时间算法用于求解最小完美支配集。
Let $G$ be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,x) is called the domination root of G. It is clear that (0,\infty) is zero free interval for domination polynomial of a graph. It is interesting to investigate graphs which have complex domination roots with positive real parts. In this paper, we first investigate complexity of the domination polynomial at specific points. Then we present and investigate some families of graphs whose complex domination ro…
In this paper, we study efficient domination in regular graphs.
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 graph is an efficient open (resp.\ closed) domination graph if there exists a subset of vertices whose open (resp.\ closed) neighborhoods partition its vertex set. Graphs that are efficient open as well as efficient closed (shortly EOCD graphs) are investigated. The structure of EOCD graphs with respect to their efficient open and efficient closed dominating sets is explained. It is shown that the decision problem regarding whether a graph is an EOCD graph is an NP-complete problem. A recursive description that constructs all EOCD trees is given and EOCD graphs are characterized among the Si…
The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown \cite{vel} that sd$(T)\leq 3$ for any tree $T$. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the \emph{domination multisubdivision number} of a nonempty graph $G$ as a minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to incr…
In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph. We also compute the domination number of some families of graphs such as star graphs, double start graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle graphs.
Let $γ(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. Rad and Volkmann posted a conjecture that $i(G)/ γ(G) \leq Δ(G)/2$ for any graph $G$, where $Δ(G)$ is its maximum degree (See \cite{5}: N.J. Rad, L. Volkmann, A note on the independent domination number in graphs. Discrete Appl. Math. 161(2013) 3087--3089). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $Δ(G)/2$ are provided as well.
In this paper, we study the notions of connected domination number and of outer-connected domination number for middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph M(G). We also compute the outer-connected domination number of some families of graphs such as star graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and some operation on graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the outer-connected domination number of middle graphs.
Let $G = (V, E)$ be a graph without an isolated vertex. A set $D\subseteq V(G)$ is a $k$-distance paired domination set of $G$ if $D$ is a $k$-distance dominating set of $G$ and the induced subgraph $\langle D \rangle$ has a perfect matching. The minimum cardinality of a $k$-distance paired dominating set for graph $G$ is the $k$-distance paired domination number, denoted by $γ_{p} ^{k}(G)$. In this paper, the $k$-distance paired domination of the flower graph $f_{n\times m}$ is discussed. For $m,n\geq 3$, the exact values for paired domination number and $2$-distance paired domination number …
Domination Index in Graphs
本研究结合图论中控制与拓扑指数概念，提出基于顶点控制度的控制指数，研究多类图的相关性质与图运算，给出相关不等式，并应用于城市设施配置，还给出找含特定顶点的极小控制集的算法。
Let G be a simple graph of order n. The domination polynomial is the generating polynomial for the number of dominating sets of G of each cardinality. A root of this polynomial is called a domination root of G. Obviously 0 is a domination root of every graph G. In the study of the domination roots of graphs, this naturally raises the question: which graphs have no nonzero real domination roots? In this paper we present some families of graphs whose have this property.
Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal …
A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$ in $G$ with $S'=S\cap V(P_m)$ can be determined via an exhaustive algorithm $Θ$ of running time $O(2^{m+n})$. Extending $Θ$ to infinite grid graphs of width $m-1$, periodicity makes the binary decision tree of $Θ$ prunable into a finite threaded tree, a closed walk of which yields all such sets $S$. The graphs induced by the complements of such sets $S$ can …
Let G be a simple graph of order n. The domination polynomial of a graph is the generating function of its dominating sets. We study the domination polynomials of generalized friendship graphs. We also consider book graphs formed by joining n copies of the cycle graph of order 4 with a common edge and study the domination polynomials of some generalized book graphs. In particular we examine the domination roots of these families and find the limiting curve for the roots.
On The Secure Domination of Mycielskian Graphs
研究Mycielskian图的 secure domination 数，给出通用界，计算重要图类的 secure domination 数，并构造实现特定 domination 参数值的图族。
In this paper we determine the signed Roman domination number of the join of cycles, wheels, fans and friendship graphs.
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,λ)=\sum_{i=0}^{n} d(G,i) λ^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,λ)$ is called the domination root of $G$. We present families of graphs whose their domination polynomial have no nonzero real roots. We observe that these graphs have complex domination roots with positive real part. Then, we consider the lexicographic product of two graphs and obtain a formula for domination polynomial of this product. Using this product, we construct a family of…
Recently, Haynes, Hedetniemi and Henning published the book Topics in Domination in Graphs, which comprises 16 contributions that present advanced topics in graph domination, featuring open problems, modern techniques, and recent results. One of these contributions is the chapter Multiple Domination, by Hansberg and Volkmann, where they put into context all relevant research results on multiple domination that have been found up to 2020. In this note, we show how to improve some results on double domination that are included in the book.
Domination and Total Domination Numbers in Zero-divisor Graphs of Commutative Rings
本文研究交换环零因子图的支配数与全支配数，证明除 $\mathbb{Z}_2 \times D$（$D$ 为整环）外二者均相等，并给出例外情形的精确值。
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.
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.
Using the domination parameters of Cayley graphs constructed out of $\mathbb{Z}_{p}\times \mathbb{Z}_{m}$, where $m\in\{p^α, p^αq^β, p^αq^βr^γ\},$ in this paper we are discussing about the total and connected domination number and diameter of these Cayley graphs.
In this paper, we determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs respectively.
The $k$-rainbow domination problem is studied for regular graphs. We prove that the $k$-rainbow domination number $γ_{rk}(G)$ of a $d$-regular graph for $d\leq k\leq 2d$ is bounded below by $\displaystyle{\left\lceil kn/2d\right\rceil}$, where $n$ is the order of a graph. We determine necessary conditions for regular graphs to attain this bound and find several examples. As an application, we determine exact $k$-rainbow domination numbers for all cubic Cayley graphs over abelian groups.
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $γ(G)$ is the domination number of $G$. In this paper we study the domination polynomials of several classes of $k$-tree related graphs. Also, we present families of these kind of graphs, whose domination polynomial have no nonzero real roots.
The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various products, ranging from explicit formulae and recurrences for specific families to more general results. As an application, we show the domination polynomial is computationally hard to evaluate.
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.
Let $G=(V,E)$ be a graph. For some $α$ with $0<α\leq 1$, a subset $S$ of $V$ is said to be a $α$-partial dominating set if $|N[S]|\geq α|V|$. The size of a smallest such $S$ is called the $α$-partial domination number and is denoted by $\mathsf{pd}_α(G)$. In this paper, we introduce $α$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $α$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
For a graph $G=(V,E)$, we call a subset $ S\subseteq V \cup E$ a total mixed dominating set of $G$ if each element of $V \cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $γ_{tm}(G)$ of $G$ is the minimum cardinality of a total mixed dominating set of $G$. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number o…
The domination multisubdivision number of a nonempty graph $G$ was defined as the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. Similarly we define the total domination multisubdivision number msd$_{γ_t}(G)$ of a graph $G$ and we show that for any connected graph $G$ of order at least two, msd$_{γ_t}(G)\leq 3.$ We show that for trees the total domination multisubdivision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for so…
A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $γ(G)$ ($Γ(G)$, respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of $G$. A subset $D\subseteq V_G$ is called a certified dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ has either zero or at least two neighbors in $V_G-D$. The cardinality of a~smallest (largest minimal, respectively) certified domi…
Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can be different, and inequalities between these five numbers are given. Finally, these fundamental dominations are interpreted in terms of the total graph of the given graph, a concept introduced by the se…
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).
In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree.
A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those sets of vertices of a graph satisfying some domination property together with other conditions on the vertices of $G$. Here, we investigate several domination related parameters in rooted product graphs.
In this paper we begin an exploration of several domination-related parameters (among which are the total, restrained, total restrained, paired, outer connected and total outer connected domination numbers) in the generalized lexicographic product (GLP for short) of graphs. We prove that for each GLP of graphs there exist several equality chains containing these parameters. Some known results on standard lexicographic product of two graphs are generalized or/and extended. We also obtain results on well $μ$-dominated GLP of graphs, where $μ$ stands for any of the above mentioned domination para…
Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.
For a graph $G$ let $γ(G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$\mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets,but $G-v$ has a unique minimum dominating set for each $v\in V(G)$. We show that each hypo-$\mathcal{UD}$ graph $G$ of order at least $3$ is connected and $γ(G-v) < γ(G)$ for all $v \in V(G)$. We obtain a …
Total outer-connected domination number of middle graphs
本文研究简单图的中间图的全外连通控制数，得出该数关于中间图阶数的紧致界，计算了特定图族的该数值，并给出了类Nordhaus-Gaddum关系。
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…
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…
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 $γ_{dR}(S(K_{n}, 2))$.
A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by $γ_{t[1,2]}(G)$. We establish two sharp upper bounds on the total [1,2]-domination number of a graph $G$ in terms of its order and minimum degree, and characterize the corresponding extremal graphs achieving these bounds. Moreover, we give some sufficient conditions for a graph without total $[1,2]$-set and for a graph with the same total $[1,2]$-domination n…
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…
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