Related papers: Prime bound of a graph

Let $G$ be a simple undirected graph. The regular number of $G$ is defined to be the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is regular. In this work, we obtain…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhoods cover all vertices of $G$, while the packing number $\rho(G)$ is the maximum size of a vertex set whose closed neighborhoods are…

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of…

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For…

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique…

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by…

For a graph $G$, a subset $S$ of $V(G)$ is a {\it hop dominating set} of $G$ if every vertex not in $S$ has a $2$-step neighbor in $S$. The {\it hop domination number}, $\gamma_h(G)$, of $G$ is the minimum cardinality of a hop dominating…

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…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…

Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by \rho(G) the set of primes which divide some character degrees of G. The prime…

A subset $S$ of vertices in a graph $G$ is a secure dominating set of $G$ if $S$ is a dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is…

Let $G=(V,E)$ be a graph and $p$ be a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set if each vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination number $\g_p(G)$ is the size of a smallest…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

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…

A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the…

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…

The basis number of a graph $G$ is the smallest integer $k$ such that $G$ admits a basis $B$ for its cycle space, where each edge of $G$ belongs to at most $k$ members of $B$. In this note, we show that every non-planar graph that can be…

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating…

The reconfiguration graph of the $k$-colorings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_{k}(G)$ if they differ in color on exactly one vertex. A graph…