Related papers: Domination structure for number three
A dominating set in a graph is a set of vertices with the property that every vertex in the graph is either in the set or adjacent to something in the set. The domination sequence of the graph is the sequence whose $k$th term is the number…
Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a…
Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which…
We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We…
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…
In this paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we present a lower bound on signed domination number of trees in terms of the order, leaves and…
A dominating set $S$ of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of $S$, and the minimum cardinality of such a set is called the…
In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the…
A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all…
For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…
A total dominating set of a graph G with no isolated vertices is a subset S of the vertex set such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of…
A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The…
A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number…
The concepts of domination and topological index hold great significance within the realm of graph theory. Therefore, it is pertinent to merge these concepts to derive the domination index of a graph. A novel concept of the domination index…
The domination multisubdivision number of a nonempty graph $G$ was defined as the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. Similarly we define…
We give an efficient construction of a reasonably small dominating set in a circulant graph on $n$ notes and $k$ distinct chord lengths. This result is based on bounds on some double exponential sums. .
A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex is adjacent to a vertex in $S$. The total domination number $\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The total…
In this paper, we study the notions of connected domination number and of outer-connected domination number for middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph M(G). We also compute the…
We investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving…
Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total domination number of $D$, denoted $\gamma_t(D)$, is the minimum cardinality among all total…