Related papers: Power domination in regular claw-free graphs
Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph…
Let $\gamma_g(G)$ be the game domination number of a graph $G$. Rall conjectured that if $G$ is a traceable graph, then $\gamma_g(G) \le \left\lceil \frac{1}{2}n(G)\right\rceil$. Our main result verifies the conjecture over the class of…
An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient…
The {\em independent domination number} $\gamma^i(G)$ of a graph $G$ is the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$. It is known \cite{abz} that in chordal graphs $\gamma^i$ is equal…
While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number…
The $3/5$-conjecture for the domination game states that the game domination numbers of an isolate-free graph $G$ on $n$ vertices are bounded as follows: $\gamma_g(G)\leq \frac{3n}5 $ and $\gamma_g'(G)\leq \frac{3n+2}5 $. Recent progress…
The complete bipartite graph $K_{1,3}$ is called a claw. The properties of claw-free graphs have attracted considerable attention, with research on claw-free planar graphs tracing back to Plummer's work in 1989. In this paper, we extend…
We use a dynamic programming algorithm to establish a new lower bound on the domination number of complete cylindrical grid graphs of the form $C_n\square P_m$, that is, the Cartesian product of a path and a cycle, when $n\equiv 2\pmod{5}$,…
The zero forcing number is the minimum number of black vertices that can turn a white graph black following a single neighbour colour forcing rule. The zero forcing number provides topological information about linear algebra on graphs,…
In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ {\em dominates} itself and its neighbors in $G$. A vertex set $D \subseteq V$ is an {\em efficient dominating set} ({\em e.d.} for short) of $G$ if every $v \in V$ is dominated in…
In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this…
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 numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$. The strength $\textrm{str}_{f}\left( G\right)$ of a numbering $f:V\left( G\right)…
In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma\_{p,k}(G-e)$,…
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.…
In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to…
In this paper, we consider the connected power domination number ($\gamma_{P, c}$) of three standard graph products. The exact value for $\gamma_{P, c}(G\circ H)$ is obtained for any two non-trivial graphs $G$ and $H.$ Further, tight upper…
Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$…
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…
In this paper, we study "robust" dominating sets of random graphs that retain the domination property even if a small \emph{deterministic} set of edges are removed. We motivate our study by illustrating with examples from wireless networks…