In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [Opuscula Math. 31 (2011), 519--531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph $G$ is a dominating set $S$ in $G$ with the property that the subgraph induced by the open neighborhood of the set $S$ has no isolated vertex. The neighborhood total domination number, denoted by $\gnt(G)$, is the minimum cardinality of a NTD-set of $G$. Every total dominating set is a NTD-set, implying that $γ(G) \le \gnt(G) \le \gt(G)$, where $γ(G)$ and $\gt(G)$ denote the domination and total domination numbers of $G$, respectively. Arumugam and Sivagnanam posed the problem of characterizing the connected graphs $G$ of order $n \ge 3$ achieving the largest possible neighborhood total domination number, namely $\gnt(G) = \lceil n/2 \rceil$. A partial solution to this problem was presented by Henning and Rad [Discrete Applied Mathematics 161 (2013), 2460--2466] who showed that $5$-cycles and subdivided stars are the only such graphs achieving equality in the bound when $n$ is odd. In this paper, we characterize the extremal trees achieving equality in the bound when $n$ is even. As a consequence of this tree characterization, a characterization of the connected graphs achieving equality in the bound when $n$ is even can be obtained noting that every spanning tree of such a graph belongs to our family of extremal trees.
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.
A locating-dominating set (LDS) of a graph $G$ is a dominating set $S$ of $G$ such that for every two vertices $u$ and $v$ in $V(G) \setminus S$, $N(u)\cap S \neq N(v)\cap S$. The locating-domination number $γ^{L}(G)$ is the minimum cardinality of a LDS of $G$. Further if $S$ is a total dominating set then $S$ is called a locating-total dominating set. In this paper we determine the domination, total domination, locating-domination and locating-total domination numbers for hypertrees and sibling trees.
For $S\subseteq V(G)$, we define $\bar{S}=V(G)\setminus S$. A set $S\subseteq V(G)$ is called a super dominating set if for every vertex $u\in \bar{S}$, there exists $v\in S$ such that $N(v)\cap \bar{S}=\{u\}$. The super domination number $γ_{sp}(G)$ of $G$ is the minimum cardinality among all super dominating sets in $G$. The super domination subdivision number $sd_{γ_{sp}}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided in order to increase the super domination number of $G$. In this paper, we investigate the ratios between super domination and other domination para…
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.
Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard). More than one guard is allowed to move in response to an attack. The $m$-eternal domination number is the minimum number of guards needed to defend the graph. We characterize the trees achieving several upper and lower bounds on the $m$-eternal domination number.
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.
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 …
Let $G$ be a simple graph with vertex set $V$ and edge set $E$, and let $S \subseteq V$. The \emph{open neighborhood} of $v \in V$, $N(v)$, is the set of vertices adjacent to $v$; the \emph{closed neighborhood} is given by $N[v] = N(v) \cup \{v\}$. The \emph{open neighborhood} of $S$, $N(S)$, is the union of the open neighborhoods of vertices in $S$, and the \emph{closed neighborhood} of $S$ is $N[S] = S \cup N(S)$. The sets $ \mathcal{P}^i(S), i \geq 0$, of vertices \emph{monitored} by $S$ at the $i^{\ {th}}$ step are given by $\mathcal{P}^0(S) = N[S]$ and $\mathcal{P}^{i+1}(S) = \mathcal{P}^…
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…
In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set $S$ of $G$ in which every vertex is totally dominated by a vertex in $S$. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game …
For a graph $G$ with domination number $γ$, Hedetniemi, Jacobs and Trevisan [European Journal of Combinatorics 53 (2016) 66-71] proved that $m_{G}[0,1)\leq γ$, where $m_{G}[0,1)$ means the number of Laplacian eigenvalues of $G$ in the interval $[0,1)$. Let $T$ be a tree with diameter $d$. In this paper, we show that $m_{T}[0,1)\geq (d+1)/3$. However, such a lower bound is false for general graphs. All trees achieving the lower bound are completely characterized. Moreover, for a tree $T$, we establish a relation between the Laplacian eigenvalues, the diameter and the domination number by showin…
一个图$G$的$k-$拟完美控制集（$k\ge 1$）是一个顶点子集$S$，使得不在$S$中的每个顶点都与$S$中的至少一个且至多k个顶点相邻.用$γ_{\stackrel{}{1k}}(G)$表示$G$中最小k-拟完全控制集的基数.这些集合最早由Chellali et al.（2013）作为完美支配概念的推广而引入。准完美支配链$γ_{\stackrel{}{11}}(G)\geγ_{\stackrel{}{12}}(G)\ge\dots\geγ_{\stackrel{}{1Δ}}(G)=γ(G)$表示当你向更完美的支配移动时，它在大小上损失了什么。我们提供了一个上界的$γ_{\stackrel{}{1k}}(T)$在任何树$T$和树实现这个界限的特点。证明了存在满足该链中所有可能的等式和不等式的树，并给出了在任意树上计算$γ_{\stackrel{}{1k}}(T)$的线性算法。
The zeroth-order general Randić index of graph $G=(V_G,E_G)$, denoted by $^0R_α(G)$, is the sum of items $(d_{v})^α$ over all vertices $v\in V_G$, where $α$ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $^0R_α$ of trees with a domination number $γ$, in intervals $α\in(-\infty,0)\cup(1,\infty)$ and $α\in(0,1)$, respectively. The corresponding extremal graphs of these bounds are also characterized.
In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the domination number. Given the sheer scale of modern networks, many existing domination type structures are expensive to implement. Variations on the theme of dominating and total dominating sets studied to date tend to focus on adding restrictions which in turn raises their implementation costs. As an alternative route a relaxation of the domination number, called disjunctive domination, was proposed and studied by Goddard et al. A set $D$ of vertices in $G$ is a disjunctive …
In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G \circ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.
Balanced Domination in Convex Polytopes, Trees, and Grid Graphs
Faculty of Natural Sciences and Mathematics, University of Banja Luka（巴尼亚卢卡大学自然科学与数学学院）；University of Nova Gorica（新戈里察大学）；Institute of Information Science (IZUM)（信息科学研究所）
For a graph $G = (V, E)$ with vertex set $V$ and edge set $E$, a subset $F$ of $E$ is called an $\emph{edge dominating set}$ (resp. a $\emph{total edge dominating set}$) if every edge in $E\backslash F$ (resp. in $E$) is adjacent to at least one edge in $F$, the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of $G$ is the {\em edge domination number} (resp. {\em total edge domination number}) of $G$, denoted by $γ^{'}(G)$ (resp. $γ_t^{'}(G)$). In the present paper, we prove that the total edge domination problem is NP-complete for bipartite graphs with maximu…
Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $γ_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\ge2$. We show that $γ_c(Q_n)\sim 2^n/n$. We use Hamming codes and an "expansion" method to construct leafy spanning trees in $Q_n$.
A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Σ_{v\in V} f(v)$. The Roman domination number, $γ_R(G)$, of $G$ is the minimum weight of an RDF on $G$. An RDF of minimum weight is called a $γ_R$-function. A graph G is said to be $γ_R$-excellent if for each vertex $x \in V$ there is a $γ_R$-function $h_x$ on $G$ with $h_x(x) \not = 0$. We present a constructive characterization of $γ_R$-excellent trees using labelings. A graph $G$ is…
The total domination number $γ_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that there exist $k$ different vertices in $G$ with the degree sum at most $m(G)$. It is conjectured that $γ_{t}(G)\leq a(G)+1$ holds for every nontrivial connected graph $G$. The conjecture has been proved for graphs with minimum degree at least $3$, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that …
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 Grundy domination number of a graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$ satisfying $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Recently, a generalization of this concept called $k$-Grundy domination was introduced. In $k$-Grundy domination, a vertex $v$ can be included in $S$ if it has a neighbor $u$ such that $u$ appears in the closed neighborhood of fewer than $k$ vertices of $S$. In this paper, we determine the $k$-Grundy domination number for some families of graphs, find degree-based bounds fo…
The Grundy domination number, ${γ_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $Γ(G)$, which is in turn greater than or equal to the independence number $α(G)$. In this paper, we initiate the study of the class of graphs $G$ with $Γ(G)={γ_{\rm gr…
A set $D\subseteq V$ of a graph $G=(V,E)$ is called a neighborhood total dominating set of $G$ if $D$ is a dominating set and the subgraph of $G$ induced by the open neighborhood of $D$ has no isolated vertex. Given a graph $G$, \textsc{Min-NTDS} is the problem of finding a neighborhood total dominating set of $G$ of minimum cardinality. The decision version of \textsc{Min-NTDS} is known to be \textsf{NP}-complete for bipartite graphs and chordal graphs. In this paper, we extend this \textsf{NP}-completeness result to undirected path graphs, chordal bipartite graphs, and planar graphs. We also…
Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textit{restrained domination number}, denoted by $γ_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. A function $f : V \rightarrow \{0, 1, 2\}$ is a \textit{restrained Italian dominating function} on $G$ if (i) for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N_G(v)} f(u) \geq 2$, (ii) the subgraph induced by $\{v \in V \mid f(v)=0 \}$ has no isolated vert…
We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.
For $c\in \mathbb{R}^{+}\cup \{\infty \}$ and a graph $G$, a function $f:V(G)\rightarrow \{0,1,c\}$ is called a $c$-self dominating function of $G$ if for every vertex $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in N_{G}(u)\}\geq 1$ where $N_{G}(u)$ is the neighborhood of $u$ in $G$. The minimum weight $w(f)=\sum _{u\in V(G)}f(u)$ of a $c$-self dominating function $f$ of $G$ is called the $c$-self domination number of $G$. The $c$-self domination concept is a common generalization of three domination-type invariants; (original) domination, total domination and Roman domination. In this paper, w…
The domination number $γ(G)$ of a graph $G$, its exponential domination number $γ_e(G)$, and its porous exponential domination number $γ_e^*(G)$ satisfy $γ_e^*(G)\leq γ_e(G)\leq γ(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $γ_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $γ_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show $γ_{e,f}^*(T)=\frac{n+2}{6}$ and $γ_e(T…
A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance $2$ of another vertex of $S$. The \emph{semitotal domination number}, $γ_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The \emph{semitotal domination multisubdivision number} of a graph $G$, $msd_{γ_{t2}}(G)$, is the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the semitotal domination number of $G$. In this paper, we show that $msd_{γ_{t2}}(G)\leq 3$ for any graph…
A set of vertices $W$ in a connected graph $G$ is called a Steiner dominating set if $W$ is both Steiner and dominating set. The Steiner domination number $γ_{st}(G)$ is the minimum cardinality of a Steiner dominating set of $G$. A linear algorithm is proposed in this paper for finding a minimum Steiner dominating set for a tree $T$.
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.
To monitor electrical activity throughout the power grid and mitigate outages, sensors known as phasor measurement units can installed. Due to implementation costs, it is desirable to minimize the number of sensors deployed while ensuring that the grid can be effectively monitored. This optimization problem motivates the graph theoretic power dominating set problem. In this paper, we propose a novel integer program for identifying minimum power dominating sets by formulating a set cover problem. This problem's constraints correspond to neighborhoods of zero forcing forts; we study their struct…
A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted by $γ_t(G)$. The total dominating set $D$ is called a total co-independent dominating set if $V(G)\setminus D$ is an independent set and has at least one vertex. The minimum cardinality of any total co-independent dominating set is denoted by $γ_{t,coi}(G)$. In this paper, we show that, for any tree $T$ of order $n$ and diameter at least three, $n-β(T)\leq …
We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.
A dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex outside $D$ is adjacent to a vertex in $D$. A locating-dominating set of $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $γ_L(G)$, is the …
A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of a dominating (respectively, total dominating) set of $D$ is the domination (respectively, total domination) number of $D$, denoted $γ(D)$ (respectively,$γ_t(D)$). The maximum number of pairwise disjoint closed (respectively,open) in-neighborhoods of $D$ is denoted by $ρ(D)$ (respectively,$ρ^{\rm o}(D)$). We prove that in digraphs whose underlying graphs hav…
The aim of this work is to investigate the nonnegative signed domination number $γ^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $γ^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Turán we bound $γ^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. …
