论文标题
关于泄漏的强迫和弹性
On leaky forcing and resilience
论文作者
论文摘要
泄漏是一个顶点，在零强迫过程中不允许执行力。 最近引入了泄漏强迫作为零强迫的新变化，以分析网络中的泄漏如何破坏零强迫过程。 $ \ ell $ -leaky强迫的图形是尽管$ \ ell $泄漏，但最小的零强迫集的大小可能会强制图形。图形$ g $是$ \ ell $ - 如果其零强制号与其$ \ ell $ -leaky迫使号相同。在本文中，我们分析了$ \ ell $ -leaky的强迫，并表明，如果$（\ ell-1）$ - 泄漏的强迫套装$ b $足够强大，那么$ b $是$ \ ell $ - $ -Leaky-leaky-leaky迫使套件。这提供了表征$ \ ell $ -leaky的强迫集的框架。此外，我们考虑了$ \ ell $ rimilient图的结构含义。我们将这些结果应用于限制$ \ ell $ leaky的强迫数量，包括树木，超三角形和网格图。特别是，我们解决了Dillman和Kenter提出的一个问题，该问题在$ 1 $ - 渗出的网格图数上的上限上提出了一个问题。
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. The $\ell$-leaky forcing number of a graph is the size of the smallest zero forcing set that can force a graph despite $\ell$ leaks. A graph $G$ is $\ell$-resilient if its zero forcing number is the same as its $\ell$-leaky forcing number. In this paper, we analyze $\ell$-leaky forcing and show that if an $(\ell-1)$-leaky forcing set $B$ is robust enough, then $B$ is an $\ell$-leaky forcing set. This provides the framework for characterizing $\ell$-leaky forcing sets. Furthermore, we consider structural implications of $\ell$-resilient graphs. We apply these results to bound the $\ell$-leaky forcing number of several graph families including trees, supertriangles, and grid graphs. In particular, we resolve a question posed by Dillman and Kenter concerning the upper bound on the $1$-leaky forcing number of grid graphs.
