Related papers: Various Characterizations of Throttling Numbers

Strong structural controllability (SSC) guarantees networked system with linear-invariant dynamics controllable for all numerical realizations of parameters. Current research has established algebraic and graph-theoretic conditions of SSC…

Lazy burning is a recently introduced variation of burning where only one set of vertices is chosen to burn in the first round. In hypergraphs, lazy burning spreads when all but one vertex in a hyperedge is burned. The lazy burning number…

A minority process in a weighted graph is a dynamically changing coloring. Each node repeatedly changes its color in order to minimize the sum of weighted conflicts with its neighbors. We study the number of steps until such a process…

In this paper, we introduce a generalization of graphlets to heterogeneous networks called typed graphlets. Informally, typed graphlets are small typed induced subgraphs. Typed graphlets generalize graphlets to rich heterogeneous networks…

Let $G$ be a simple graph. The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated…

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can…

In 2018, forts were defined as non-empty subsets of vertices in a graph where no vertex outside the set has exactly one neighbor in the set. Forts have since been used to characterize zero forcing sets, model zero forcing as an integer…

For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…

Graph burning is a discrete-time process that models the propagation of information in a network. Initially, we have an undirected graph of unburned vertices. At each time step, an unburned vertex is chosen to burn; additionally, unburned…

The forcing number of a perfect matching $M$ in a graph $G$ is the smallest number of edges inside $M$ that can not be contained in other perfect matchings. The anti-forcing number of $M$ is the smallest number of edges outside $M$ whose…

Stabilization of graphs has received substantial attention in recent years due to its connection to game theory. Stable graphs are exactly the graphs inducing a matching game with non-empty core. They are also the graphs that induce a…

For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring…

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In…

The foldings of a connected graph $G$ are defined as follows. First, $G$ is a folding of itself. Let $G'$ be a graph obtained from $G$ by identifying two vertices at distance 2 in $G$. Then every folding of $G'$ is a folding of $G$. The…

Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number,…

A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower…

Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which…

We investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color {\alpha} is…

The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…