标题： 连通强制密度及相关问题 显示英文标题

标题： Connected forcing density and related problems

摘要： 一个连通强制集是诱导连通子图的零强制集。 在本文中，我们引入并研究CF稠密图——所有顶点都属于某个最小连通强制集的图。 我们确定了几类CF稠密图，并研究了CF稠密性与零强制和全强制中类似概念之间的关系。 我们还对CF稠密树进行了表征，并给出了树中不同连通强制集数量的公式。 最后，我们分析了CF稠密性在笛卡尔乘积、联图和冠图等图运算下是否被保留。 显示英文摘要

摘要： A connected forcing set of a graph is a zero forcing set that induces a connected subgraph. In this paper, we introduce and study CF-dense graphs -- graphs in which every vertex belongs to some minimum connected forcing set. We identify several CF-dense graph families and investigate the relationships between CF-density and analogous notions in zero forcing and total forcing. We also characterize CF-dense trees and give a formula for the number of distinct connected forcing sets in trees. Finally, we analyze when CF-density is preserved under graph operations such as Cartesian products, joins, and coronas.