Related papers: Domination number of middle graphs
In this paper, we provide a new upper bound for the alpha-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another…
We study the problem of half-domination sets of vertices in vertex transitive infinite graphs generated by regular or semi-regular tessellations of the plane. In some cases, the results obtained are sharp and in the rest, we show upper…
This papers focuses on the average order of dominating sets of a graph. We find the extremal graphs for the maximum and minimum value over all graphs on $n$ vertices, while for trees we prove that the star minimizes the average order of…
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…
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…
A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in…
In this paper, we study the inverse signed total domination number in graphs and present new lower and upper bounds on this parameter. For example by making use of the classic theorem of Turan (1941), we present a sharp upper bound for…
The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by…
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…
Let $G=(V,E)$ be a graph and $p$ a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set of $G$ if every vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination number is the minimum cardinality of a…
The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from…
A dominating set of a graph is a set of vertices such that every vertex not in the set has at least one neighbor in the set. The problem of counting dominating sets is #P-complete for chordal graphs but solvable in polynomial time for its…
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda)=\sum_{i=0}^{n} d(G,i) \lambda^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,\lambda)$ is…
In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the…
The total domination number of a graph $G$ without isolated vertices is the minimum number of vertices that dominate all vertices in $G$. The total bondage number $b_t(G)$ of $G$ is the minimum number of edges whose removal enlarges the…
The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…
Given a connected graph with domination (or total domination) number \gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of dominating and total dominating sets of size \gamma. An exact answer is provided for \gamma=2and…
Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and…
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 $R$…
We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph.…