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 classes of graphs, as well as finding general bounds and characterizations. Also, we find some results on the upper $k$-tuple total domination number of the Cartesian and cross product graphs.
In a graph $G$, a vertex dominates itself and its neighbours. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $D$ dominates every vertex of $G$ at least $k$ times. The minimum cardinality among all $k$-tuple dominating sets is the $k$-tuple domination number of $G$. In this paper, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case $k=2$. In addition, we improve two well-known lower bounds on the $k$-tuple domination number.
For $k \ge 1$ an integer, a set $S$ of vertices in a graph $G$ with minimum degree at least~$k-1$ is a $k$-tuple dominating set of $G$ if every vertex of $S$ is adjacent to at least $k-1$ vertices in $S$ and every vertex of $V(G) \setminus S$ is adjacent to at least $k$ vertices in $S$; that is, $|N_G[v] \cap S| \ge k$ for every vertex $v$ of $G$ where $N_G[v]$ denotes the closed neighborhood of $v$ which consists of $v$ and all neighbors of $v$. A $k$-tuple restrained dominating set of $G$ is a $k$-tuple dominating set $S$ of $G$ with the additional property that every vertex outside $S$ has …
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…
Let $G$ be a connected graph of order $n$, whose minimum vertex degree is at least $k$. A subset $S$ of vertices in $G$ is a $k$-tuple total dominating set if every vertex of $G$ is adjacent to at least $k$ vertices in $S$. The minimum cardinality of a $k$-tuple total dominating set of $G$ is the $k$-tuple total domination number of $G$, denoted by $γ_{\times k,t}(G)$. Henning and Yeo in \cite{hen} proved that if $G$ is a cubic graph different from the Heawood graph, $γ_{\times 2, t}(G) \leq \frac{5}{6}n$, and this bound is sharp. Similarly, a $k$-tuple dominating set is a subset $S$ of vertic…
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 called the $k$-tuple total dominating number and it is denoted by $γ_{\times k,t}(G)$. We give a constructive proof of a general formula for $γ_{\times 3, t}(K_n \Box K_m)$.
The $k$-tuple domination problem, for a fixed positive integer $k$, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least $k$ vertices in this set. The $k$-tuple domination is NP-hard even for chordal graphs. For the class of circular-arc graphs, its complexity remains open for $k\geq 2$. A $0,1$-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In thi…
Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$ vertices of $V(G)-S$. The minimum number of vertices of a such sets in $G$ are the $k$-tuple total restrained domination number $γ_{\times k,t}^{r}(G)$ of $G$. The maximum number of classes of a partition of $V(G)$ such that its all classes are $k$-tuple total restrained dominating sets in $G$, is called the $k$-tuple total restrained domatic number of $G$. In t…
The inflated graph $G_{I}$ of a graph $G$ with $n(G)$ vertices is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $u\in X_{i}$, $v\in X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. For integer $k\geq 1$, the $k$-tuple total domination number $γ_{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$, which is a set of vertices in $G$ such that every verte…
We consider two general frameworks for multiple domination, which are called <r,s>-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the <r,s>-domination and parametric domination numbers. These generalisations imply new upper bounds for the {k}-domination and total k-domination numbers. Also, we study threshold functions, which impose additional restrictions on the minimum vertex degree, and present new upper bounds for the a…
For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$. The minimum cardinality of a $k$-tuple dominating set of $G$ is the $k$-tuple domination number of $G$. When $k=1$, a $k$-tuple domination number is the well-studied domination number. We define the $k$-tuple domatic number of $G$ as the largest number of sets in a partition of $V$ into $k$-tuple dominating sets. Recall that when $k=1$, a $k$-tuple domatic nu…
In combinatorics, a latin square is a $n\times n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square graph. In this article, we compute lower and upper bounds for the domination number and the k-tuple total domination numbers of such graphs. Moreover, we describe a formula for the 2-tuple total domination number.
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 vertex in a graph dominates itself and each of its adjacent vertices. The \emph{$k$-tuple domination problem}, for a fixed positive integer $k$, is to find a minimum sized vertex subset in a given graph such that every vertex is dominated by at least $k$ vertices of this set. From the computational point of view, this problem is NP-hard. It follows from previous works by Bui-Xuan et al.~(2013) and by Belmonte et al.~(2013) -- in the context of locally checkable vertex subset problems in graph classes with quickly computable and bounded min-width -- that the $k$-tuple domination problem is so…
In this article, we study the $d$-distance $m$-tuple ($\ell, r$)-domination problem. Given a simple undirected graph $G=(V, E)$, and positive integers $d, m, \ell$ and $r$, a subset $V' \subseteq V$ is said to be a $d$-distance $m$-tuple ($\ell, r$)-dominating set if it satisfies the following conditions: (i) each vertex $v \in V$ is $d$-distance dominated by at least $m$ vertices in $V'$, and (ii) each $r$ size subset $U$ of $V$ is $d$-distance dominated by at least $\ell$ vertices in $V'$. Here, a vertex $v$ is $d$-distance dominated by another vertex $u$ means the shortest path distance bet…
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…
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.
An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we give a characterization of 3-arc graphs and obtain sharp upper bounds on the domination number of the 3-arc graph of a graph $G$ in terms that of $G$.
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 broadcast on a graph $G=(V,E)$ is a function $f:V \rightarrow \{0,1, \ldots, \text{diam}(G)\}$ satisfying $f(v) \leq e(v)$ for all $v \in V$, where $e(v)$ denotes the eccentricity of $v$ and $\text{diam}(G)$ denotes the diameter of $G$. We say that a broadcast dominates $G$ if every vertex can hear at least one broadcasting node. The upper domination number is the maximum cost of all possible minimal broadcasts, where the cost of a broadcast is defined as $\text{cost} (f)= \sum_{v \in V}f(v)$. In this paper we establish both the upper domination number and the upper broadcast domination numb…
For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the number of end-vertices and penultimate vertices of $G$, respectively. We show that $γ_{\times2}(G)\geq s\ell_{\times2}(G)$, where $γ_{\times2}(G)$ is the well-known double domination number of a graph $G$ with no isolated vertices. We prove that the problem of deciding whether the equality holds for a given graph is NP-complete even when restricted to $4$-parti…
It is conjectured that the game domination number is at most $3n/5$ for every $n$-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the class of forests and that of graphs with minimum degree at least two. Here we prove that the slightly bigger upper bound $5n/8$ is valid for every isolate-free graph.
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…
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…
Generalizing work on graphs, Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs, attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\geq 7$. Furthermore, we show that the squid hyp…
In his 1992 Ph.D. thesis Chang identified an efficient way to dominate $m \times n$ grid graphs and conjectured that his construction gives the most efficient dominating sets for relatively large grids. In 2011 Gonçalves, Pinlou, Rao, and Thomassé proved Chang's conjecture, establishing a closed formula for the domination number of a grid. In March 2013 Fata, Smith and Sundaram established upper bounds for the $k$-distance domination numbers of grid graphs by generalizing Chang's construction of dominating sets to $k$-distance dominating sets. In this paper we improve the upper bounds establis…
The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement must be connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.
In this paper, we provide a new upper bound for the alpha-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the alpha-rate domination number, which combines the concepts of alpha-domination and k-tuple domination.
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.
An algorithmic upper bound on the domination number $γ$ of graphs in terms of the order $n$ and the minimum degree $δ$ is proved. It is demonstrated that the bound improves best previous bounds for any $5\le δ\le 50$. In particular, for $δ=5$, Xing et al.\ proved in 2006 that $γ\le 5n/14 < 0.3572 n$. This bound is improved to $0.3440 n$. For $δ=6$, Clark et al.\ in 1998 established $γ<0.3377 n$, while Biró et al. recently improved it to $γ<0.3340 n$. Here the bound is further improved to $γ< 0.3159 n$. For $δ=7$, the best earlier bound $0.3 088 n$ is improved to $γ< 0.2927 n$.
Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $γ_{t}$($G$), is the minimum cardinality of a total dominating set of $G$. In this paper using a greedy algorithm we provide an upper bound for $γ_{t}$($G$), whenever $G$ is a bipartite graph and $δ(G)$ $\geq$ $k$. More precisely, we show that if $k$ > 1 is a natural number, then for every bipartite graph $G$ of order $n$ and $δ(G) \ge k$, $ $$γ_{t}$($G$) $\leq$ $n(1- \frac{k!}{\prod_{i=0}^{k-1}(\frac{k}{k-1}+i)}).$
Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$
Faculty of Mechanical Engineering, University of Ljubljana（卢布尔雅那大学机械工程学院）；Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那数学、物理与力学研究所）；Rudolfovo – Science and Technology Centre, Novo Mesto（诺沃梅斯托鲁多尔福沃科学与技术中心）
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, we study the inverse signed total domination number in graphs and present new lower and upper bounds on this parameter. For example by making use of the classic theorem of Turan (1941), we present a sharp upper bound for graphs with no induced complete subgraph of order greater than two. Also, we bound this parameter for a tree in terms of its order and the number of leaves and characterize all trees attaining this bound.
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.
We first introduce the concept of (k,k',k'')-domination numbers in graphs, which is a generalization of many domination parameters. Then we find lower and upper bounds for this parameter, which improve many well-known results in literatures.
Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at least $b$ vertices in $S$. The minimum cardinality of an $(a,b)$-dominating set of $G$ is the $(a,b)$-domination number of $G$, denoted by $γ_{a,b}(G)$. There are various results about upper bounds for $γ_{a,b}(G)$ when $G$ is regular or $a$ and $b$ are small numbers. In the first part of this paper, for a given graph $G$ with the minimum degree of $\max\{a,b…
We study the Nordhaus-Gaddum type results for $(k-1,k,j)$ and $k$-domination numbers of a graph $G$ and investigate these bounds for the $k$-limited packing and $k$-total limited packing numbers in graphs. As the special case $(k-1,k,j)=(1,2,0)$ we give an upper bound on $dd(G)+dd(\overline{G})$ stronger than that presented by Harary and Haynes (1996). Moreover, we establish upper bounds on the sum and product of packing and open packing numbers and characterize all graphs attaining these bounds.
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…
In this paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we present a lower bound on signed domination number of trees in terms of the order, leaves and support vertices.
In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this parameter for graphs with no complete graph of order r+1 as a subgraph. Also, we prove that n-2(s-s') is an upper bound on the signed total domination number of any tree of order n with s support vertices and s' support vertives of degree two. Moreover, we characterize all trees attainig this bound.
We prove that for every graph $G$ on $n$ vertices and with minimum degree five, the domination number $γ(G)$ cannot exceed $n/3$. The proof combines an algorithmic approach and the discharging method. Using the same technique, we provide a shorter proof for the known upper bound $4n/11$ on the domination number of graphs of minimum degree four.
After $2$-crossing-critical graphs were characterized in 2016, their most general subfamily, large $3$-connected $2$-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their domination and independence number.
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.
The first three authors [European J. Combin. 33 (2012), 62--71] established a relationship between the transversal number and the domination number of uniform hypergraphs. In this paper, we establish a relationship between the total transversal number and the total domination number of uniform hypergraphs. We prove tight asymptotic upper bounds on the total transversal number in terms of the number of vertices, the number of edges, and the edge size.
Erdős [On Schütte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $\log (n+1)$ vertices, where $\log$ is the logarithm to base $2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $\log n - 2 \log \log n + 1$. This notion of directed domination number has been generalized to arbitrary graphs by Caro and Henning in [Directed domination in oriented graphs, Discrete Appl. Math. (2012) 160:7--8.]. However, the generalization to directed r-uniform hypergraphs seems…
