Related papers: Total 2-domination of proper interval graphs

With the aid of hypergraph transversals it is proved that $\gamma_t(Q_{n+1}) = 2\gamma(Q_n)$, where $\gamma_t(G)$ and $\gamma(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the…

We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed…

The domination multisubdivision number of a nonempty graph $G$ was defined as the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. Similarly we define…

Given a graph $G$, a set $F$ of edges is an edge dominating set if all edges in $G$ are either in $F$ or adjacent to an edge in $F$. $G$ is said to be well-edge-dominated if every minimal edge dominating set is also minimum. In 2022, it was…

Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$…

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient…

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (e.d.s.\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for…

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 $G$,…

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 $G$, $\gamma(G)$, is the minimum order of a dominating set. A subset $R$…

The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a…

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…

The {\em independent domination number} $\gamma^i(G)$ of a graph $G$ is the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$. It is known \cite{abz} that in chordal graphs $\gamma^i$ is equal…

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of…

An eternal dominating set of a graph $G$ is a set of vertices (or "guards") which dominates $G$ and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current…

A subset $S$ of vertices in a graph $G=(V, E)$ is a Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside…

The total domination number $\gamma_{t}(G)$ 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…

An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…

The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph $G$. Each vertex chosen must strictly increase the number of vertices…

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on…