Related papers: Efficient and Perfect domination on circular-arc g…

Let $G=(V,E)$ be a graph. For an edge $e=xy\in E$, the closed neighbourhood of $e$, denoted by $N_G[e]$ or $N_G[xy]$, is the set $N_G[x]\cup N_G[y]$. A vertex set $L\subseteq V$ is liar's vertex-edge dominating set of a graph $G=(V,E)$ if…

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…

The enhanced power graph of a group G is a graph with vertex set G, where two distinct vertices x and y are adjacent if and only if there exists an element w in G such that both x and y are powers of w. To obtain the proper enhanced power…

The $k$ red domination problem for a bipartite graph $G=(X,Y,E)$ is to find a subset $D \subseteq X$ of cardinality at most $k$ that dominates vertices of $Y$. The decision version of this problem is NP-complete for general bipartite graphs…

In this paper, we study efficient domination in regular graphs.

A set $S$ of vertices in a graph $G = (V, E)$ is called {\em cycle independent} if the induced subgraph $\langle S\rangle$ is acyclic, and called {\em odd-cycle indepdendet} if $\langle S\rangle$ is bipartite. A set $S$ is {\em cycle…

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex…

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$.…

A set $S \subseteq V$ is a dominating set in G if for every u \in V \ S, there exists $v \in S$ such that $(u,v) \in E$, i.e., $N[S] = V$. A dominating set $S$ is an Isolate Dominating Set} (IDS) if the induced subgraph $G[S]$ has at least…

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs.…

For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired…

We consider two graph optimization problems called vector domination and total vector domination. In vector domination one seeks a small subset S of vertices of a graph such that any vertex outside S has a prescribed number of neighbors in…

Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such…

The \emph{Dominating $H$-Pattern} problem generalizes the classical $k$-Dominating Set problem: for a fixed \emph{pattern} $H$ and a given graph $G$, the goal is to find an induced subgraph $S$ of $G$ such that (1) $S$ is isomorphic to $H$,…

For a graph $G=(V,E)$, a set $S \subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v \in V \setminus S$ is dominated by at most two vertices of $S$, i.e. $1 \leq \vert N(v) \cap S \vert \leq 2$. Moreover a set…

A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the…

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…

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper,…

Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real…