相关论文: The zero forcing span of a graph

In this paper, we study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with…

We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of $\text{pw}(G)+1$, where $\text{pw}(G)$ is the pathwidth of $G$. Starting from a path decomposition, the algorithm runs in $O(nm)$ time,…

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…

The burning and forcing processes are both instances of propagation processes on graphs that are commonly used to model real-world spreading phenomena. The contribution of this paper is two-fold. We first establish a connection between…

The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For any perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M such…

Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the…

Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white,…

The forcing number of a perfect matching $M$ of a graph $G$ is the smallest cardinality of subsets of $M$ that are contained in no other perfect matchings of $G$. The forcing spectrum of $G$ is the collection of forcing numbers of all…

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…

Let $G=(V,E)$ be a finite connected graph along with a coloring of the vertices of $G$ using the colors in a given set $X$. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in…

Zero forcing is a combinatorial game played on a graph with the ultimate goal of changing the colour of all the vertices at minimal cost. Originally this game was conceived as a one player game, but later a two-player version was devised…

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…

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup…

The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a…

The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\setminusS$ are colored white) such that $V(G)$ is turned black after finitely many applications of…

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 this paper we begin the study of well-failed graphs, that is, graphs in which every maximal failed zero forcing set is a maximum failed zero forcing set, or equivalently, in which every minimal fort is a minimum fort. We characterize…

A graph in which all minimal zero forcing sets are in fact minimum size is called ``well-forced." This paper characterizes well-forced trees and presents an algorithm for determining which trees are well-forced. Additionally, we…

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in…

Zero forcing parameters, associated with graphs, have been studied for over a decade, and have gained popularity as the number of related applications grows. In particular, it is well-known that such parameters are related to certain vertex…