完全多部图和图积上的不可逆k-阈值和多数转换过程
Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products
摘要：
在疾病通过人群传播、观点通过社交网络传播以及故障通过分布式计算机网络传播的图论模型中，顶点处于两种状态，黑色或白色，并且这些状态根据模型中使用的特定转换过程的规则在离散时间步长动态更新。本文考虑了不可逆k-阈值过程和多数转换过程。在不可逆的k阈值中（分别为，大多数）转换过程中，如果至少k（相应地，至少有一半的邻居是黑人。k转换集（相应地，动态垄断）是一组顶点，如果其最初被着色为黑色，则将导致所有顶点最终在k阈值以下被着色为黑色（分别，大部分）的转换过程。我们回答了几个公开的问题，提出的界限和一些精确值的最小顶点数的k-转换集和动态垄断的完全多部图，以及笛卡尔和张量积的两个图。
英文摘要：
In graph theoretical models of the spread of disease through populations, the spread of opinion through social networks, and the spread of faults through distributed computer networks, vertices are in two states, either black or white, and these states are dynamically updated at discrete time steps according to the rules of the particular conversion process used in the model. This paper considers the irreversible k-threshold and majority conversion processes. In an irreversible k-threshold (resp., majority) conversion process, a vertex is permanently colored black in a certain time period if at least k (resp., at least half) of its neighbors were black in the previous time period. A k-conversion set (resp., dynamic monopoly) is a set of vertices which, if initially colored black, will result in all vertices eventually being colored black under a k-threshold (resp., majority) conversion process. We answer several open problems by presenting bounds and some exact values of the minimum number of vertices in k-conversion sets and dynamic monopolies of complete multipartite graphs, as well as of Cartesian and tensor products of two graphs.
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