Related papers: On the Relationships between Zero Forcing Numbers …
We consider three probability measures on subsets of edges of a given finite graph $G$, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the…
Probabilistic zero forcing is a graph coloring process in which blue vertices "infect" (color blue) white vertices with a probability proportional to the number of neighboring blue vertices. We introduce reversion probabilistic zero forcing…
In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430]. The throttling number $th_c(G)$ of a graph $G$ is the minimum possible value of $k…
In this paper, we initiate the study of the inverse eigenvalue problem for probe graphs. A probe graph is a graph whose vertices are partitioned into probe vertices and non-probe vertices such that the non-probe vertices form an independent…
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among…
The decycling number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. It is known that determining the decycling number of a graph $G$ is equivalent to finding the maximum induced…
The maximum oriented $k$-forcing number of a simple graph $G$, written $\MOF_k(G)$, is the maximum directed $k$-forcing number among all orientations of $G$. This invariant was recently introduced by Caro, Davila and Pepper in…
A global forcing set for maximal matchings of a graph $G=(V(G), E(G))$ is a set $S \subseteq E(G)$ such that $M_1\cap S \neq M_2 \cap S$ for each pair of maximal matchings $M_1$ and $M_2$ of $G$. The smallest such set is called a minimum…
We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges of a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ of $G$ is said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \sum_{e\in E(H)} f(e)…
An identifying code of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists…
Reconfiguration graphs provide a way to represent relationships among solutions to a problem, and have been studied in many contexts. We investigate the reconfiguration graphs corresponding to minimum PSD forcing sets and minimum skew…
In a graph G, a dominating set Df subset of V (G) is called a dom-forcing set if the sub-graph induced by Df must form a zero forcing set. The minimum cardinality of such a set is known as the dom-forcing number of the graph G, denoted by…
Product throttling answers the question of minimizing the product of the resources needed to accomplish a task, and the time in which it takes to accomplish the task. In product throttling for positive semidefinite zero forcing, task that…
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 basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis…
A leak is a vertex that is not allowed to perform a force during the zero forcing process. Leaky forcing was recently introduced as a new variation of zero forcing in order to analyze how leaks in a network disrupt the zero forcing process.…
Given a graph $G$, one may ask: "What sets of eigenvalues are possible over all weighted adjacency matrices of $G$?" (The weight of an edge is positive or negative, while the diagonal entries can be any real numbers.) This is known as the…
We define the cover number of a graph $G$ by a graph class $\mathcal P$ as the minimum number of graphs of class $\mathcal P$ required to cover the edge set of $G$. Taking inspiration from a paper by Harary, Hsu and Miller, we find an exact…
The minimum status of a graph is the minimum of statuses of all vertices of this graph. We give a sharp upper bound for the minimum status of a connected graph with fixed order and matching number (domination number, respectively), and…
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with…