Related papers: Italian domination in generalized Petersen graphs

In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The Inverse Domination Conjecture says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with…

A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$, denoted by $\gamma _t (G)$, is the minimum cardinality of a…

Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number,…

This paper delves into the stability of the $2$-domination number in simple undirected graphs. The $2$-domination number of a graph $G$, $\gamma_2(G)$, represents the minimum size of a vertex subset where every other vertex in the graph is…

A total Roman dominating function (TRDF) on a graph $G$ with no isolated vertices is a function $f:V(G)\to\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has a neighbor assigned $2$, and the subgraph induced by $\{v:f(v)>0\}$ has no…

Given a graph $G = (V,E)$ and two its distinct vertices $u$ and $v$. The $(u,v)$-$P_k$-{\em addition graph} of $G$ is the graph $G_{u,v,k-2}$ obtained from disjoint union of $G$ and a path $P_k: x_0,x_1,..,x_{k-1}$, $k \geq 2$, by…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$.The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The Grundy domination number, ${\gamma_{\rm gr}}(G)$, 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…

We commence the study of domination in the incidence graphs of combinatorial designs. Let $D$ be a combinatorial design and denote by $\gamma(D)$ the domination number of the incidence (Levy) graph of $D$. We obtain a number of results…

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is a…

A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $\iota(G,F)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$…

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…

In a graph $G$, a set $D\subseteq V(G)$ is called 2-dominating set if each vertex not in $D$ has at least two neighbors in $D$. The 2-domination number $\gamma_2(G)$ is the minimum cardinality of such a set $D$. We give a method for the…

Given a partition ${\mathcal V}=(V_1, \ldots,V_m)$ of the vertex set of a graph $G$, an {\em independent transversal} (IT) is an independent set in $G$ that contains one vertex from each $V_i$. A {\em fractional IT} is a non-negative real…

In the Roman domination problem, an undirected simple graph $G(V,E)$ is given. The objective of Roman domination problem is to find a function $f:V\rightarrow {\{0,1,2\}}$ such that for any vertex $v\in V$ with $f(v)=0$ must be adjacent to…

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by…

Let $\gamma(G)$ be the domination number of a graph $G$. A graph $G$ is \emph{domination-vertex-critical}, or \emph{$\gamma$-vertex-critical}, if $\gamma(G-v)< \gamma(G)$ for every vertex $v \in V(G)$. In this paper, we show that: Let $G$…

We investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving…

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a…

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…