Related papers: Averaging $2$-Rainbow Domination and Roman Dominat…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner…

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the…

A graph is reconstructible if it is determined up to isomorphism from the collection of all its one-vertex-deleted subgraphs, known as the deck of G. The Reconstruction Conjecture (RC) posits that every finite simple graph with at least…

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…

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating…

In this paper, we study a parameter that is squeezed between arguably the two important domination parameters, namely the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a…

For a simple graph $G$, denote by $n$, $\Delta(G)$, and $\chi'(G)$ its order, maximum degree, and chromatic index, respectively. A connected class 2 graph $G$ is edge-chromatic critical if $\chi'(G-e)<\Delta(G)+1$ for every edge $e$ of $G$.…

A dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex outside $D$ is adjacent to a vertex in $D$. A locating-dominating set of $G$ is a dominating set $D$ of $G$ with the additional property that every two…

The domination number of a graph $G$, denoted by $\gamma (G)$, is the minimum cardinality of a dominating set of $G$. A vertex of a graph is called critical if its deletion decreases the domination number, and a graph is called critical if…

Let $G=(V,E)$ be a finite connected simple graph with vertex set $V$ and edge set $E$. A signed Roman dominating function (SRDF) on a graph $G$ is a function $f: V \rightarrow \{-1, 1, 2\}$ that satisfies two conditions: (i) $\sum_{y\in…

For a graph $G$, a function $f:V(G) \to \{0,1,2\}$ is called a $2$-limited dominating broadcast on $G$ if for every vertex $u$, there exists a vertex $v$ such that $f(v)>0$ and the distance between $u$ and $v$ in $G$ is at most $f(v)$. The…

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…

A dominating set of a graph $G$ is a subset $D \subseteq V_G$ 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 domination number…

Let $G$ be a simple and finite graph without isolated vertices. In this paper we introduce and study a new degree sequence derived invariant called the \emph{sub-total domination number}, denoted $\sub_t(G)$. In particular, we show that…

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 $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In…

We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order $n$ with fixed domination…

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

A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $\gamma_{st}(G)$ of $G$ is the…

Let $\gamma(G)$ and $\beta(G)$ denote the domination number and the covering number of a graph $G$, respectively. A connected non-trivial graph $G$ is said to be $\gamma\beta$-{perfect} if $\gamma(H)=\beta(H)$ for every non-trivial induced…

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is…