Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number $γ_k(G)$ is the minimum cardinality of a $k$-dominating set in $G$. For any graph $G$, we know that $γ_k(G) \geq γ(G)+k-2$ where $ Δ(G)\geq k\geq 2$ and this bound is sharp for every $k\geq 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\geq 3$ and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when $k=3$. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.
A subset $D\subseteq V(G)$ is called a $k$-distance dominating set of $G$ if every vertex in $V(G)\setminus D$ is within distance $k$ from some vertex of $D$. The minimum cardinality among all $k$-distance dominating sets of $G$ is called the $k$-distance domination number of $G$. In this note we give upper bound on the $k$-distance domination number of a connected bipartite graph and improve some results have been given like Theorem 2.1 and 2,7 in [Tian and Xu, A note on distance domination of graphs, Australian Journal of Combinatorics, 43 (2009), 181-190].
A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The \emph{location-domination number} of $G$, $λ(G)$, is the minimum cardinality of a locating-dominating set. In this work we study relationships between $λ({G})$ and $λ(\overline{G})$ for bipartite graphs. The main result is the characterization of all connected bipartite graphs $G$ satisfying $λ(\overline{G})=λ({G})+1$. To this aim, we define an edge-labeled graph $G^S$ as…
A subset $D$ of the vertex set $V$ of a graph $G$ is called an $[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the minimum number of vertices is called a $γ_{[1,k]}$-set and the number of its vertices is the $[1,k]$-domination number $γ_{[1,k]}(G)$ of $G$. In this short note we show that the decision problem whether $γ_{[1,k]}(G)=n$ is an $NP$-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph $G$ of order $n$ satisfying $γ_{[1,k]}(G)=n$ is given for every …
A sequence $(v_1,\ldots ,v_k)$ of vertices in a graph $G$ without isolated vertices is called a total dominating sequence if every vertex $v_i$ in the sequence totally dominates at least one vertex that was not totally dominated by $\{v_1,\ldots , v_{i-1}\}$ and $\{v_1,\ldots ,v_k\}$ is a total dominating set of $G$. The length of a shortest such sequence is the total domination number of G ($γ_t(G)$), while the length of a longest such sequence is the Grundy total domination number of $G$ ($γ_{gr}^t(G)$). In this paper we study graphs with equal total and Grundy total domination number. We ch…
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 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.
Recently the notion of $k$-rainbow total domination was introduced for a graph $G$, motivated by a desire to reduce the problem of computing the total domination number of the generalized prism $G \Box K_k$ to an integer labeling problem on $G$. In this paper we further demonstrate usefulness of the labeling approach, presenting bounds on the rainbow total domination number in terms of the total domination number, the rainbow domination number and the rainbow total domination number, as well as the usual domination number, where the latter presents a generalization of a result by Goddard and H…
For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision …
Consider all $k$-element subsets and $\ell$-element subsets $(k>\ell )$ of an $n$-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding $\ell$-element set is a subset of the corresponding $k$-element set. Let $G_{k,\ell}$ denote this graph. The domination number of $G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value ($n$ tending to infinity) of the domination number of $G_{k,2}$. Here we prove the conjecture, determining the asymptotic value of the domination number $γ(G_{k,2})={k+3\…
Let ${[n] \choose k}$ and ${[n] \choose l}$ $( k > l ) $ where $[n] = \{1,2,3,...,n\}$ denote the family of all $k$-element subsets and $l$-element subsets of $[n]$ respectively. Define a bipartite graph $G_{k,l} = ({[n] \choose k},{[n] \choose l},E)$ such that two vertices $S\, ε\,{[n] \choose k} $ and $T\, ε\,{[n] \choose l} $ are adjacent if and only if $T \subset S$. In this paper, we give an upper bound for the domination number of graph $G_{k,2}$ for $k > \lceil \frac{n}{2} \rceil$ and exact value for $k=n-1$.
Dominating sets and resolving sets have important applications in control theory and computer science. In this paper, we introduce an edge-analog of the classical dominant metric dimension of graphs. By combining the concepts of a vertex-edge dominating set and an edge resolving set, we introduce the notion of a vertex-edge dominant edge resolving set of a graph. We call the minimum cardinality of such a set in a graph $\G$, the vertex-edge dominant edge metric dimension $\g_{emd}(\G)$ of $\G$. The new parameter $\g_{emd}$ is calculated for some common families such as paths, cycles, complete …
A dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. A set $C\subseteq V_G$ is a covering set of $G$ if every edge of $G$ has at least one vertex in $C$. The covering number $β(G)$ of $G$ is the minimum cardinality of a covering set of $G$. The set of connected graphs $G$ for which $γ(G)=β(G)$ is denoted by ${\cal C}_{γ=β}$, while ${\cal B}$ denotes the set of all connected bipartite graphs in which the domination numbe…
Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super dominating set of $G$ is the super domination number of $G$. An exact formula for the super domination number of a tree $T$ is obtained and demonstrated that a smallest super dominating set of $T$ can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph $G$ is at most a given integer if $G$ is a b…
A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The domination number, $γ(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. For an even integer $n\ge2$ and $1\leΔ\le\lfloor\log_2n\rfloor$, a Knödel graph $W_{Δ,n}$ is a $Delta$-regular bipartite graph of even order $n$, with vertices$(i,j)$, for $i=1,2$ and $0\le j\le n/2-1$, where for every $j$, $0\le j\le n/2-1$, there is an edge between $(1,j)$ and $(2,j+2^k-1 \text{(mod(n/2)})$, for $k=0,1,\cdots,Δ-1$. In this paper, we determine the do…
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…
A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $γ(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is called domination vertex critical, or just $γ$-critical if removal of any vertex decreases the domination number. A graph $G$ is called domination vertex stable, or just $γ$-stable, if removal of any vertex does not decrease the domination number. For an even integer $n\ge 2$ and $1\le Δ\le \lfloor \log_2n \rfloor$, a Knödel graph $W_{Δ,n}$ is a $Δ$-regular bipa…
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. …
The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($γ_{\rm tcg}(G)$) $γ_{\rm cg}(G)$ of $G$. We show that $γ_{\rm tcg}(G)\in \{γ_{\rm cg}(G), γ_{\rm cg}(G) + 1, γ_{\rm cg}(G) + 2\}$, and consequently define $G$ as …
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…
Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duchêne, Gledel, Parreau, and Renault. Both games are instances of the combinatorial Maker-Breaker games. The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately take turns choosing vertices of $G$. The first player, Dominator, selects a vertex in order to totally dominate $G$ while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. It is shown that there are infinit…
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.
The $2$-domination number $γ_2(G)$ of a graph $G$ is the minimum cardinality of a set $S\subseteq V(G)$ such that every vertex from $V(G)\setminus S$ is adjacent to at least two vertices in $S$. The annihilation number $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $G$ is at most the number of its edges. It was conjectured that $γ_2(G) \leq a(G) +1$ holds for every connected graph $G$. The conjecture was earlier confirmed, in particular, for graphs of minimum degree $3$, for trees, and for block graphs. In this paper, we disp…
An open neighbourhood locating-dominating set is a set $S$ of vertices of a graph $G$ such that each vertex of $G$ has a neighbour in $S$, and for any two vertices $u,v$ of $G$, there is at least one vertex in $S$ that is a neighbour of exactly one of $u$ and $v$. We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of vertices. More precisely, we prove that these graphs are exactly the graphs all whose connected components are half-graphs (a half-graph is a special bipartite graph with both parts of the same size, where each part can be ordered s…
A dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In this paper we provide a new characterization of bipartite graphs whose domination number is equal to the cardinality of its smaller partite set. Our characterization is based upon a new graph operation.
