Related papers: On the Unimodality of Domination Polynomials

A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a…

Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by {\gamma}pr(G), is the size of…

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 $R$…

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we…

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 $\gamma_g(G)$ and $\gamma_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $\gamma_g$-perfect (resp. $\gamma_{tg}$-perfect), if every induced subgraph $F$ of $G$…

A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is…

For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive alliance if for each vertex $v$ in $V-S$ at least half the vertices from 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…

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $\gamma(G)$ is the domination…

Let $G$ be a graph without isolated vertices. The total domination number of $G$ is the minimum number of vertices that can dominate all vertices in $G$, and the paired domination number of $G$ is the minimum number of vertices in a…

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…

Let $G$ be a connected graph. A non-empty $T\subseteq V(G)$ is a $2$-\textit{movable total dominating set} of $G$ if $T$ is a total dominating set and for every pair $x,y \in T$, $T \backslash \{x, y\}$ is a total dominating set in $G$, or…

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…

A semitotal dominating set of a graph $G$ with no isolated vertex is a dominating set $D$ of $G$ such that every vertex in $D$ is within distance two of another vertex in $D$. The minimum size $\gamma_{t2}(G)$ of a semitotal dominating set…

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…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

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…

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a…

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…