Related papers: Lower Boundary Independent Broadcasts in Trees

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag…

A fair dominating set in a graph $G$ (or FD-set) is a dominating set $S$ such that all vertices not in $S$ are dominated by the same number of vertices from $S$; that is, every two vertices not in $S$ have the same number of neighbors in…

The Min Energy broadcast problem consists in assigning transmission ranges to the nodes of an ad-hoc network in order to guarantee a directed spanning tree from a given source node and, at the same time, to minimize the energy consumption…

Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the weight of $f$ is $w(f)=\sum_{u\in V}f(u)$ which is just the total number of pebbles…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$.…

A radio labeling of a graph $G$ is a function $f : V(G) \rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the…

The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has…

We consider the corner-stone broadcast task with an adaptive adversary that controls a fixed number of $t$ edges in the input communication graph. In this model, the adversary sees the entire communication in the network and the random…

The burning number $b(G)$ of a graph $G$ is the minimum number of rounds required to burn all vertices when, at each discrete step, existing fires spread to neighboring vertices and one new fire may be ignited at an unburned vertex. This…

Consider a graph $G$ and a real-valued function $f$ defined on the degree set of $G$. The sum of the outputs $f(d_v)$ over all vertices $v\in V(G)$ of $G$ is usually known as the vertex-degree-function indices and is denoted by $H_f(G)$,…

The leap eccentric connectivity index of $G$ is defined as $$L\xi^{C}(G)=\sum_{v\in V(G)}d_{2}(v|G)e(v|G)$$ where $d_{2}(v|G) $ be the second degree of the vertex $v$ and $e(v|G)$ be the eccentricity of the vertex $v$ in $G$. In this paper,…

Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\zeta(v)$ as $\zeta(v)={\max}_{H: v\in H}~\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show…

Domination in graphs has long been studied and is applied to signal distribution problem. For example, telecommunication companies want to spread the signal from broadcast stations by transmitting varying signal strength to all receiving…

Given a graph and a subset of its nodes, referred to as source nodes, the minimum broadcast problem asks for the minimum number of steps in which a signal can be transmitted from the sources to all other nodes in the graph. In each step,…

Given a graph $G=(V,E)$, the problem of \gb{} is to find a sequence of nodes from $V$, called burning sequence, in order to burn the whole graph. This is a discrete-step process, in each step an unburned vertex is selected as an agent to…

A set $S\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices…

The $k$-independence number of a graph $G$ is the maximum size of a set of vertices at pairwise distance greater than $k$. In this paper, for each positive integer $k$, we prove sharp upper bounds for the $k$-independence number in an…

A signed edge domination function (or SEDF) of a simple graph $G=(V,E)$ is a function $f: E\rightarrow \{1,-1\}$ such that $\sum_{e'\in N[e]}f(e')\ge 1$ holds for each edge $e\in E$, where $N[e]$ is the set of edges in $G$ that share at…

Message broadcasting in networks could be carried over spanning trees. A set of spanning trees in the same network is node independent if two conditions are satisfied. First, all trees are rooted at node $r$. Second, for every node $u$ in…

For each integer k >= 2, we determine a sharp bound on mad(G) such that V(G) can be partitioned into sets I and F_k, where I is an independent set and G[F_k] is a forest in which each component has at most k vertices. For each k we…