Title: The $q$-Analogue of Zero Forcing for Certain Families of Graphs Show Chinese title

Title: $q$的零强迫类似物对于某些图族的分析

Abstract: 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-conjunction with studies on the inertia of a graph, and has become known as the $q$-analogue of zero forcing. In this paper, we study and compute the $q$-analogue zero forcing number for various families of graphs. We begin with by considering a concept of contraction associated with trees. We then significantly generalize an equation between this $q$-analogue of zero forcing and a corresponding nullity parameter for all threshold graphs. We close by studying the $q$-analogue of zero forcing for certain Kneser graphs, and a variety of cartesian products of structured graphs. Show Chinese abstract

Abstract: 零 forcing 是一种在图上进行的组合游戏，最终目标是以最小的成本改变所有顶点的颜色。 最初这个游戏被设想为单人游戏，但后来在研究图的惯性时，设计了一个双人版本，并被称为$q$的零 forcing 模仿。 在本文中，我们研究并计算了各种图族的$q$模仿零 forcing 数。 我们首先考虑与树相关的一个收缩概念。 然后，我们显著地推广了一个方程，该方程将这种$q$模仿零 forcing 与所有阈值图的相应零空间参数联系起来。 最后，我们研究了某些 Kneser 图的$q$模仿零 forcing，以及多种结构图的笛卡尔积。