Let $X_{n}$ denote the unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$. We continue the study of cases in which the inequality $γ_t(X_n) \le g(n)$ is strict, where $γ_t$ denotes the total domination number, and $g$ is the arithmetic function known as Jacobsthal's function. The best that is currently known in this direction is a construction of Burcroff which gives a family of $n$ with arbitrarily many prime factors that satisfy $γ_t(X_n) \le g(n)-2$. We present a new interpretation of the problem which allows us to use recent results on the computation of Jacobsthal's function to construct $n$ with arbitrarily many prime factors that satisfy $γ_t(X_n) \le g(n)-16$. We also present new lower bounds on the domination numbers of direct products of complete graphs, which in turn allow us to derive new asymptotic lower bounds on $γ(X_n)$, where $γ$ denotes the domination number. Finally, resolving a question of Defant and Iyer, we completely classify all graphs $G = \prod_{i=1}^t K_{n_i}$ satisfying $γ(G) = t+2$.
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…
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.
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…
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…
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.
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.
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.
Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time.
The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete r-partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour, Akbari and Ebrahimi.
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 $i$. We consider the lexicographic product of two specific graphs and study their domination polynomials.
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…
The product power throttling number of a graph is defined to study product throttling for power domination. The domination number of a graph is an upper bound for its product power throttling number. It is established that the two parameters are equal for certain families including paths, cycles, complete graphs, unit interval graphs, and grid graphs (on the plane, cylinder, and torus). Families of graphs for which the product power throttling number is less than the domination number are also exhibited. Graphs with extremely high or low product power throttling number are characterized and bo…
Let $G=(V,E)$ be a graph and $p$ a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set of $G$ if every vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination number is the minimum cardinality of a $p$-dominating set in $G$. In this paper, we establish an exact formula of the $p$-domination number of all complete multipartite graphs for arbitrary positive integer $p$.
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 $G$.
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…
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…
The unitary Cayley graph of $\mathbb{Z} /n \mathbb{Z}$, denoted $X_{\mathbb{Z} / n \mathbb{Z}}$, has vertices $0,1, \dots, n-1$ with $x$ adjacent to $y$ if $x-y$ is relatively prime to $n$. We present results on the tightness of the known inequality $γ(X_{\mathbb{Z} / n \mathbb{Z}})\leq γ_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)$, where $γ$ and $γ_t$ denote the domination number and total domination number, respectively, and $g$ is the arithmetic function known as Jacobsthal's function. In particular, we construct integers $n$ with arbitrarily many distinct prime factors such that $γ(X_{\math…
The connected domination game was recently introduced by Borowiecki, Fiedorowicz and Sidorowicz as another variation of the domination game. The rules are essentially the same, except that the set of played vertices must be connected at all stages of the game. We answer a problem from their paper regarding the relation between the number of moves in a game where Dominator/Staller starts the game. In this paper we also study the relation to the diameter and present graphs with small connected game domination number. We determine the values on the lexicographic product graphs, and consider the e…
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.
In a graph $G$, a vertex dominates itself and its neighbours. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The minimum cardinality among all double dominating sets of $G$ is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs $G\circ H$ in terms of invariants of the factor graphs $G$ and $H$.
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…
The closed neighborhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having an end-vertex in common with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1,1\}$. If $\sum_{x\in{N[e]}}f(x)\geq 1$ for each edge $e \in E(G)$, then $f$ is called a signed edge dominating function of $G$. The signed edge domination number of $G$ is the minimum weight of a signed edge dominating function of $G$. In this paper, we find the signed edge domination number of the complete tripartite graph $K_{m,n,p}$, where $1\leq m\leq n$ and $p\geq m+n$.…
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…
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 $G$ 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…
Let k be a positive integer and let f be a map from V(G) to the set of all subsets of {1,2,3,...,k}. The function f is called a k-rainbow dominating function of G provided that whenever u is a vertex of G such that f(u) is the empty set, then for each integer r in {1,2,3,...,k} there is a neighbor x of u such that f(x) contains r. The k-rainbow domination number of G is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function of G. The k-rainbow domination number of G is the ordinary domination number of the Cartesian product …
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…
The Grundy domination number of a simple graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$, $v_i \in V$, that satisfies the property $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Here, $N(v) = \{u : uv \in E\}$ and $N[v] = N(v) \cup \{v\}$. In this note, we prove a recent conjecture about the Grundy domination number of the strong product of two graphs. We then discuss how this result relates to the zero forcing number of the strong product of graphs.
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.
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.
In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.
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.
In \cite{nr-1996} Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that $i(G\times H) \ge i(G)i(H)$ where $i(G)$ is the independent domination number of $G$ and $G\times H$ is the direct product of graphs $G$ and $H$. We show this conjecture is false, and, in fact, construct pairs of graphs for which $\min\{i(G), i(H)\} - i(G\times H)$ is arbitrarily large. We also give the exact value of $i(G\times K_n)$ when $G$ is either a path or a cycle.
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.
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…
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…
A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $γ_{gr}^{t}(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $γ_{gr}^t(G\times H) \geq γ_{gr}^t(G)γ_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic produ…
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{}
The first multiplicative Zagreb index $Π_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index $Π_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for $Π_1$ and upper bound for $Π_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds.
The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ 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 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 dominating number of lexicographic product graphs in terms of invariants of the factor graphs…
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…
A set $D$ of vertices in a graph $G$ is called dominating if every vertex of $G$ is either in $D$ or adjacent to a vertex of $D$. The paired domination number $γ_{\mathrm{pr}}(G)$ of $G$ is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number $Γ(G)$ is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of two multiplicative inequalities for these domination parameters, where the graph product is the direct product $\times$. We show that for every positive constant $c$, there exist graphs $G…
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 $G$ be a graph on $n$ vertices and $m$ edges and $D(G,x)$ the domination polynomial of $G$. In this paper we completely characterize the values of $n$ and $m$ for which optimal graphs exist for domination polynomials. We also show that there does not always exist least optimal graphs for the domination polynomial. Applications to network reliability are highlighted.
In this paper, we proceed on determining the minimum $q_{min}$ among the connected nonbipartite graphs on $n\geq 5$ vertices and with domination number $\frac{n+1}{3}<γ\leq \frac{n-1}{2}$. Further results obtained are as follows: $\mathrm{(i)}$ among all nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{n-1}{2}$, the minimum $q_{min}$ is completely determined; $\mathrm{(ii)}$ among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{o}\leq5$ and domination number at least $\frac{n+1}{3}<γ\leq \frac{n-2}{2}$, the minimum $q_{min}$ is completely determi…
For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index $\prod_2$ is equal to the product of the products of degrees of pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajstić in 1972. In this paper, we explore the mutiplicative Zagreb indices in terms of arbitrary domination number. The sharp upper and lower bounds of $\prod_1(G)$ and $\prod_2(G)$ are given. In addition,…
The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a family of infinitely many integers $n$ such that $γ_t(X_n) \leq g(n) - 2$, where $γ_t$ denotes the total domination number and $g$ denotes the Jacobsthal function. We determine the irredundance number, domination number, and lower independence number of certain direct products of complete graphs and give bounds for these parameters for any direct product of compl…
A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of domination sets of each cardinality in $G$, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode $ \lceil \frac{n}{2}\rceil $.
A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the stro…
