Related papers: Disjoint dominating and 2-dominating sets in graph…

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…

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is…

Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set $C$ of a graph $G$ which is also…

Given a graph $G$, a dominating set $D$ is a set of vertices such that any vertex in $G$ has at least one neighbor (or possibly itself) in $D$. A ${k}$-dominating multiset $D_k$ is a multiset of vertices such that any vertex in $G$ has at…

We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the $k$-colouring graph $\mathcal{C}_{k}(G)$ of a graph $G$ correspond to the proper $k$-colourings of a graph $G$, with two…

Let ${\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\cal G})$ to be the…

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

The {\em disjointness graph} of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph $G$ of any system of segments in the plane is {\em…

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in…

Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…

The domination polynomial of a graph $G$ of order $n$ 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 number of $G$. The…

A D2CS of a graph G is a set $S \subseteq V(G)$ with $diam(G[S]) \leq 2$. We study the problem of counting and enumerating D2CS of a graph. First we give an explicit formula for the number of D2CS in a complete k-ary tree, Fibonacci tree,…

A graph is an efficient open (resp.\ closed) domination graph if there exists a subset of vertices whose open (resp.\ closed) neighborhoods partition its vertex set. Graphs that are efficient open as well as efficient closed (shortly EOCD…

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…

Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this…

The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the…

Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super…

The domination polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for…

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides…

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and…