Related papers: Lower Boundary Independent Broadcasts in Trees

A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in $S$ has a neighbor which is independent, or is itself independent, from all previous vertices in $S$, is called a Grundy dominating sequence. The Grundy domination…

We study the fault-tolerance of networks from both the structural and computational point of view using the minimum leaf number of the corresponding graph $G$, i.e. the minimum number of leaves of the spanning trees of $G$, and its…

A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices,…

A split-by-edges tree of a graph G on n vertices is a binary tree T where the root = V(G), every leaf is an independent set in G, and for every other node N in T with children L and R there is a pair of vertices {u, v} in N such that L = N…

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…

We investigate two recently introduced graph parameters, both of which measure the complexity of the tree decompositions of a given graph. Recall that the treewidth ${\rm tw}(G)$ of a graph $G$ measures the largest number of vertices…

For a simple graph $G=(V,E),$ let $\mathcal{S}_+(G)$ denote the set of real positive semidefinite matrices $A=(a_{ij})$ such that $a_{ij}\neq 0$ if $\{i,j\}\in E$ and $a_{ij}=0$ if $\{i,j\}\notin E$. The maximum positive semidefinite…

In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index…

We study the single-message broadcast problem in dynamic radio networks. We show that the time complexity of the problem depends on the amount of stability and connectivity of the dynamic network topology and on the adaptiveness of the…

A mobile agent, starting from a node $s$ of a simple undirected connected graph $G=(V,E)$, has to explore all nodes and edges of $G$ using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows…

Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum,…

Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the…

Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ of $G$ is the maximum of the average distances…

We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n \lambda_{2} / (\Delta + \lambda_{2}) - 1$, where $\Delta$ and $\lambda_{2}$ are the maximum degree and the second smallest…

Let $G$ be a finite, connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ and the proximity $\pi(G)$ of $G$ are the maximum and…

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance from $v$ to any other vertex. The vertices whose eccentricity are equal to the diameter (the maximum eccentricity) of $G$ are called peripheral vertices. In trees the…

We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$…

A subgraph H= (V, F) of a graph G= (V,E) is non-separating if G-F, that is, the graph obtained from G by deleting the edges in F, is connected. Analogously we say that a subdigraph X= (V,B) of a digraph D= (V,A) is non-separating if D-B is…

In a directed graph $G$ with non-correlated edge lengths and costs, the \emph{network design problem with bounded distances} asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria…

A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into independent sets or color classes. A vertex v Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj . A coloring is a Grundy…