Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $...

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps. Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explo...

Zero forcing (also called graph infection) on a simple, undirected graph G is based on the colorchange rule: If each vertex of G is colored either white or black, and vertex v is a black vertex with only one white neighbor w, then change the color of w to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color cha...

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the pap...

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the pap...

Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored under a specific color change rule. The goal is to find a minimum set of vertices such that after iteratively applying the rule, all of the vertices become colored (i.e., a minimum zero forcing set). Of particular in...

Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as to model the spread of diseases and information in social networks. In this paper, we introduce and study the connected forcing process – a restriction of z...

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures C of the same type, there exist B 6= A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka’s Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardi...

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to א1. Later we give applications, among them the c...

Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the entire graph to blue is called a zero forcing ...