Related papers: Trees with Maximum p-Reinforcement Number

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…

We prove that for every graph $G$ on $n$ vertices and with minimum degree five, the domination number $\gamma(G)$ cannot exceed $n/3$. The proof combines an algorithmic approach and the discharging method. Using the same technique, we…

For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set…

Given a graph $G = (V, E)$, a set $S \subseteq V \cup E$ of vertices and edges is called a mixed dominating set if every vertex and edge that is not included in $S$ happens to be adjacent or incident to a member of $S$. The mixed domination…

Given a positive integer $k$ and graph $G$, the $\mathbb{Z}_k$-Ramsey number $R(G,\mathbb{Z}_k)$ is the least $N$ (if it exists) such that every coloring $f:E(K_N)\rightarrow \mathbb{Z}_k$ contains a copy $G'$ of $G$ such that $\sum_{e\in…

A dominating set in a graph $G$ is a subset of vertices $D$ such that every vertex in $V\setminus D$ is a neighbor of some vertex of $D$. The domination number of $G$ is the minimum size of a dominating set of $G$ and it is denoted by…

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 graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number…

In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the…

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from…

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as…

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…

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…

A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is paired if the subgraph induced by its vertices has a perfect matching, and…

In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular,…

For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $\gamma_p(G),$ is the minimum cardinality of a perfect…

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erdos-Renyi random graph G(n,p). For all p(n) in [0,1], we show that, with high probability, T is precisely the minimum between delta and…

In a graph $G$, a vertex dominates itself and its neighbors. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The double domination number $\gamma_{\times 2}(G)$ is…

Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k \subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of $G$ is called the $k$-nearly edge independence number of $G$, and is denoted by…