A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $γ(G)$, is the minimum order of a dominating set. A subset $R$ of edges of a graph $G$ is a 2-degree-packing, if any three edges from $R$ do not have the same incident vertex. The 2-degree-packing number of $G$, $ν_2(G)$, is the maximum order of a 2-degree-packing of $G$.
In this paper, we prove that any simple graph $G$ satisfies $γ(G)\leqν_2(G)-1$. Furthermore, we give a characterization of simple connected graphs $G$ satisfying $γ(G)=ν_2(G)-1$.
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.
Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We pr…
Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $S\subseteq V(G)$, we define the external neighbourhood of $S$ as the set $N_e(S)$ of all vertices in $V(G)\setminus S$ having at least one neighbour in $S$. The differential of $S$ is defined to be $\partial(S)=|N_e(S)|-|S|$. In this paper, we introduce the study of the $2$-packing differential of a graph, which we define as $\partial_{2p}(G)=\max\{\partial(S): S\subseteq V(G) \text{ is a }2\text{-packing}\}.$ We show that the $2$-packing differential is closely related to several graph parameters, including the packing num…
A set $S\subseteq V$ is $α$-dominating if for all $v\in V-S$, $|N(v) \cap S | \geq α|N(v)|.$ The $α$-domination number of $G$ equals the minimum cardinality of an $α$-dominating set $S$ in $G$. Since being introduced by Dunbar, et al. in 2000, $α$-domination has been studied for various graphs and a variety of bounds have been developed. In this paper, we propose a new parameter derived by flipping the inequality in the definition of $α$-domination. We say a set $S \subset V$ is a $β$-packing set of a graph $G$ if $S$ is a proper, maximal set having the property that for all vertices $v \in V-…
We give a sharp lower bound on the lower $k$-limited packing number of a general graph. Moreover, we establish a Nordhaus-Gaddum type bound on $2$-limited packing number of a graph. Also, we investigate the concepts of packing number ($1$-limited packing number) and open packing number in graphs with more details. In this way, by making use of the classic result of Meir and Moon (1975) and its total version (2005) we give new sharp upper bounds on domination number and total domination number of trees.
A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along wit…
Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $γ(G)$, $γ_{\rm pr}(G)$, and $γ_{t}(G)$, respectively. For a positive integer $k$, a $k$-packing in $G$ is a set $S \subseteq V(G)$ such that for every pair of distinct vertices $u$ and $v$ in $S$, the distance between $u$ and $v$ is at least $k+1$. The $k$-packing number is the order of a largest $k$-packing and is denoted by $ρ_{k}(G)$. It is well known that $γ_{\rm pr}(G) \le 2γ(G)$. In this paper, we prove that it is NP-hard t…
The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence, respectively. In 2001, Erwin introduced \emph{broadcast domination} in graphs, a covering problem using balls of various radii, where the cost of a ball is its radius. The minimum cost of a dominating broadcast in a graph $G$ is denoted by $γ_b(G)$. The dual (in the sense of linear programming) of broadcast domination is \emph{multipacking}: a multipack…
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'…
In this paper, we find upper bounds on the open packing and $k$-limited packing numbers with emphasis on the cases $k=1$ and $k=2$. We solve the problem of characterizing all connected graphs on $n$ vertices with $ρ_{o}(G)=n/δ(G)$ which was presented in 2015 by Hamid and Saravanakumar. Also, by establishing a relation between the $k$-limited packing number and double domination number we improve two upper bounds given by Chellali and Haynes in 2005.
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…
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.
A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$ such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset $D$ of points of a linear system $(P,\mathcal{L})$ is a dominating set of $(P,\mathcal{L})$ if for every $u\in P\setminus D$ there exists $v\in D$ such that $u,v\in l$, for some $l\in\mathcal{L}$. The cardinality of a mini…
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…
For a sequence $d$ of non-negative integers, let ${\cal F}(d)$ be the set of all forests whose degree sequence is $d$. We present closed formulas for $γ_{\max}^{\cal F}(d)=\max\{ γ(F):F\in {\cal F}(d)\}$ and $α_{\min}^{\cal F}(d)=\min\{ α(F):F\in {\cal F}(d)\}$ where $γ(F)$ and $α(F)$ are the domination number and the independence number of a forest $F$, respectively.
Let $G=(V,E)$ be a graph of order $n$ and let $γ_{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $δ\geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $γ_{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k…
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 $γ< 0.2927 n$.
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $Γ(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$…
A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number, $γ^{d}_2(G)$, of $G$ is the minimum cardinality of a disjunctive dominating set in $G$. In this paper, we show that if $G$ be a graph of order at least $3$, $δ(G)\geq 2$ and with no component isomorphic to any of eight forbidden graphs, then $γ^{d}_2(G)\leq \frac{|G|}{3}$. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also stud…
An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3…
In the domination game, introduced by Brešar, Klavžar and Rall in 2010, Dominator and Staller alternately select a vertex of a graph $G$. A move is legal if the selected vertex $v$ dominates at least one new vertex -- that is, if we have a $u\in N[v]$ for which no vertex from $N[u]$ was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the number of vertices selected. The goal of Dominator is to minimize whilst that of Staller is to maximize the length of the game. The game domination number $γ_g(G)$ of $G$ is the length of the domin…
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.
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 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 …
An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $Δ=4$ or $Δ\ge 6$, every connected $n$-vertex graph of maximum degree at most $Δ$ has an independent dominating set of size at most $(1-\fracΔ{\lfloorΔ^2/4\rfloor+Δ})(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\fracΔ{ \lfloorΔ^2/4\rfloor+Δ})n$.
Let $γ(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. In 1977, Hedetniemi and Mitchell began with the comparison of of $i(G)$ and $γ(G)$ and recently Rad and Volkmann posted a conjecture that $i(G)/ γ(G) \leq Δ(G)/2$, where $Δ(G)$ is the maximum degree of $G$. In this work, we prove the conjecture for trees and provide the graph achieved the sharp bound.
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.
Let $G$ be a simple and finite graph without isolated vertices. In this paper we introduce and study a new degree sequence derived invariant called the \emph{sub-total domination number}, denoted $\sub_t(G)$. In particular, we show that $\sub_t(G)$ serves as a lower bound on $γ_t(G)$, where $γ_t(G)$ denotes the heavily studied \emph{total domination number} of $G$.
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…
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,…
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.
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）
For a sequence $d$ of non-negative integers, let ${\cal G}(d)$ and ${\cal F}(d)$ be the sets of all graphs and forests with degree sequence $d$, respectively. Let $γ_{\min}(d)=\min\{ γ(G):G\in {\cal G}(d)\}$, $α_{\max}(d)=\max\{ α(G):G\in {\cal G}(d)\}$, $γ_{\min}^{\cal F}(d)=\min\{ γ(F):F\in {\cal F}(d)\}$, and $α_{\max}^{\cal F}(d)=\max\{ α(F):F\in {\cal F}(d)\}$ where $γ(G)$ is the domination number and $α(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $α_{\max}(d)$ can be determined in polynomial time. We establish the existence…
