Related papers: General bounds on limited broadcast domination

The $(t,r)$ broadcast domination number of a graph $G$, $\gamma_{t,r}(G)$, is a generalization of the domination number of a graph. $\gamma_{t,r}(G)$ is the minimal number of towers needed, placed on vertices of $G$, each transmitting a…

Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where $dist(T,v)$ denotes the distance between…

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…

Given a graph and an integer $k$, it is an NP-complete problem to decide whether there is a dominating set of size at most $k$. In this paper we study this problem for the Kn\"odel Graph on $n$ vertices using elementary number theory…

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…

For a graph $ G = (V, E) $ with a vertex set $ V $ and an edge set $ E $, a function $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each vertex $ u \in V $, if there exists a vertex $ v $ in $…

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph $G=(V,E)$ called the $(t,r)$ broadcast domination number which depends on the positive integer parameters $t$ and $r$. In this setting, a…

In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and…

We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph $D$ is a function $f:V(D)\to\mathbb{N}$ such that for each vertex…

Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\rightarrow \{-1,1\}$ is called a \emph{signed $k$-dominating function} of $G$ if $\sum_{u\in N_G[v]}f(u)\geq k$ for all $v\in V$. The…

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For…

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…

We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order $n$ with fixed domination…

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 k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with…

Capacitated Domination generalizes the classic Dominating Set problem by specifying for each vertex a required demand and an available capacity for covering demand in its closed neighborhood. The objective is to find a minimum-sized set of…

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…

{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph $G$, called the sub-$k$-domination number and denoted $sub_k(G)$. We show that $sub_k(G)$ is a computationally efficient sharp lower…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…