论文标题
最大失败的零强迫集针对两个图的产品
Maximal failed zero forcing sets for products of two graphs
论文作者
论文摘要
让$ g $为一个简单的有限图，带有顶点套装$ v（g）$和edge set $ e（g）$，每个顶点均为彩色蓝色或白色。使用以下颜色变化规则在$ g $上定义标准的零强迫过程：让$ s $是$ g $的所有最初蓝色顶点的集合，让$ u \在s $中。如果$ v $是$ u $ in $ g $中的唯一白色顶点，颜色$ v $ blue和更新$ s $，将$ v $添加到$ s $中。如果$ s = v（g）$在有限的颜色变化规则的迭代次数之后，我们说$ s $是$ g $的零强迫设置。否则，我们说$ s $是失败的零强迫集。在本文中，我们为诸如笛卡尔产品，强制品，词典产物和冠状的图形产品构建了最大失败的强迫集。特别是，我们考虑了两个路径，两个循环和两个完整图的产物。
Let $G$ be a simple, finite graph with vertex set $V(G)$ and edge set $E(G)$, where each vertex is either colored blue or white. Define the standard zero forcing process on $G$ with the following color-change rule: let $S$ be the set of all initially blue vertices of $G$ and let $u \in S$. If $v$ is the unique white vertex adjacent to $u$ in $G$, color $v$ blue and update $S$ by adding $v$ to $S$. If $S = V(G)$ after a finite number of iterations of the color-change rule, we say that $S$ is a zero forcing set for $G$. Otherwise, we say that $S$ is a failed zero forcing set. In this paper, we construct maximal failed zero forcing sets for graph products such as Cartesian products, strong products, lexicographic products, and coronas. In particular, we consider products of two paths, two cycles, and two complete graphs.
