Related papers: Counting independent dominating sets in linear pol…
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…
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse…
The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A…
A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…
Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial…
An independent coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$ neither of which is an independent dominating set but whose union $V_1 \cup V_2$ is an independent dominating set. An independent coalition…
Let G be a graph. The independence-domination number is the maximum over all independent sets I in G of the minimal number of vertices needed to dominate I. In this paper we investigate the computational complexity of independence…
Let $G=(V,E)$ be a simple graph. A set $S\subseteq V(G)$ is called an outer-connected dominating set (or ocd-set) of $G$, if $S$ is a dominating set of $G$ and either $S=V(G)$ or $V\backslash S$ is a connected graph. In this paper we…
Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph…
Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…
An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum…
An independent set in a graph G is a set of vertices no two of which are joined by an edge. A vertex-weighted graph associates a weight with every vertex in the graph. A vertex-weighted graph G is called a unique independence…
Let $G$ be a graph. A set $S \subseteq V(G)$ is independent if its elements are pairwise non-adjacent. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G) \setminus N[v]$ there exists $u \in N(v)$ such that $S…
A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…
Given a simple, finite, nonempty graph $G=(V(G),E(G))$, a vertex subset $D\subseteq V(G)$ is said to be a dominating set if every vertex $v\in V(G)-D$ is adjacent to a vertex in $D$. The independent domination number $\gamma_i(G)$ is the…
A set $D$ of vertices of a simple graph $G=(V,E)$ is a strong dominating set, if for every vertex $x\in \overline{D}=V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…
We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated,…
Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…
A connected dominating set in a graph is a dominating set of vertices that induces a connected subgraph. Following analogous studies in the literature related to independent sets, dominating sets, and total dominating sets, we study in this…
A power dominating set of a graph is a set of vertices that observes every vertex in the graph by combining classical domination with an iterative propagation process arising from electrical circuit theory. In this paper, we study the power…