Related papers: Efficient (j,k)-Domination in Regular Graphs

In this paper, we give a series of couterexamples to negate a conjecture and hence answer an open question on the $k$-power domination of regular graphs (see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]).…

The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal…

Let $G=(V(G),E(G))$ be a simple connected and undirected graph with vertex set $V(G)$ and edge set $E(G)$. A set $S \subseteq V(G)$ is a $dominating$ $set$ if for each $v \in V(G)$ either $v \in S$ or $v$ is adjacent to some $w \in S$. That…

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)…

A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The domination number, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. For…

A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to \{0,1,2\}$ such that for every vertex $f(v) = 0$, it holds that $\sum_{u \in N(v)} f(u) = 2$, i.e., the weight of the labels assigned by $f$ to the…

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire…

In a graph $G = (V,E)$, a k-ruling set $S$ is one in which all vertices $V$ \ $S$ are at most $k$ distance from $S$. Finding a minimum k-ruling set is intrinsically linked to the minimum dominating set problem and maximal independent set…

Let $\gamma(G)$ denote the domination number of a graph $G$. A {\it Roman domination function} of a graph $G$ is a function $f: V\to\{0,1,2\}$ such that every vertex with 0 has a neighbor with 2. The {\it Roman domination number}…

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…

In this paper, we introduce new concepts of domination and packing functions in graphs, which generalize, respectively, the labelled dominating and packing functions defined by Lee and Chang in 2008, and Hinrichsen et al. in 2019. These…

A graph is \emph{well-dominated} if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show…

We consider a generic algorithmic paradigm that we call progressive exploration, which can be used to develop simple and efficient parameterized graph algorithms. We identify two model-theoretic properties that lead to efficient progressive…

Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance-$k$ domination, guards initially occupy the vertices of a distance-$k$ dominating set. After a vertex…

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open…

A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x \in V. The…

In a simple connected graph $G=(V,E)$, a subset of vertices $S \subseteq V$ is a dominating set if any vertex $v \in V\setminus S$ is adjacent to some vertex $x$ from this subset. A number of real-life problems can be modeled using this…

Let $k$ be a positive integer. A {\em Roman $k$-dominating function} on a graph $G$ is a labeling $f:V (G)\longrightarrow \{0, 1, 2\}$ such that every vertex with label 0 has at least $k$ neighbors with label 2. A set…

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which…

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 $\gamma\_{p,k}(G-e)$,…