标题： IP模型用于最小零 forcing 集、fords 和相关图参数 显示英文标题

标题： IP Models for Minimum Zero Forcing Sets, Forts, and Related Graph Parameters

摘要： 零 forcing 是一种在图上的二进制着色游戏，其中一组已填充的顶点可以通过颜色变化规则强制非填充的顶点变为填充状态。 2008 年，证明了图的零 forcing 数是其最大零空间的上界。 此外，零 forcing 数的组合优化问题被证明是 NP 难的。 从那时起，对零 forcing 及其相关参数的研究受到了相当多的关注。 2018 年，定义了图的 forts 为非空的顶点子集，其中集合外的任何顶点在集合中恰好有一个邻居。 Forts 被用来将零 forcing 建模为整数规划并提供零 forcing 数的下界。 到目前为止，已经为图的零 forcing 数开发了三种整数规划模型：感染模型、时间步模型和 forts 覆盖模型。 在本文中，我们提出了这些模型的变体，用于计算零 forcing 数及相关图参数，如最小和最大传播时间、抑制数和分数零 forcing 数。 此外，我们还提出了几种新的模型来计算实现的传播时间区间、图的所有最小 forts 和图的 forts 数。 我们最后进行了几个数值实验，展示了当应用于小阶和中阶图时，我们的模型的有效性。 此外，我们提供了针对传播时间区间、最小 forts 数量、forts 数量以及图的分数零 forcing 数的几个开放猜想的实验证据。 显示英文摘要

摘要： Zero forcing is a binary coloring game on a graph where a set of filled vertices can force non-filled vertices to become filled following a color change rule. In 2008, the zero forcing number of a graph was shown to be an upper bound on its maximum nullity. In addition, the combinatorial optimization problem for the zero forcing number was shown to be NP-hard. Since then, the study of zero forcing and its related parameters has received considerable attention. In 2018, the forts of a graph were defined as non-empty subsets of vertices where no vertex outside the set has exactly one neighbor in the set. Forts have been used to model zero forcing as an integer program and provide lower bounds on the zero forcing number. To date, three integer programming models have been developed for the zero forcing number of a graph: the Infection Model, Time Step Model, and Fort Cover Model. In this article, we present variations of these models for computing the zero forcing number and related graph parameters, such as the minimum and maximum propagation times, throttling number, and fractional zero forcing number. In addition, we present several new models for computing the realized propagation time interval, all minimal forts of a graph, and the fort number of a graph. We conclude with several numerical experiments that demonstrate the effectiveness of our models when applied to small and medium order graphs. Moreover, we provide experimental evidence for several open conjectures regarding the propagation time interval, the number of minimal forts, the fort number, and the fractional zero forcing number of a graph.