正则图上两层多数规则下小联盟的力量
The Power of Small Coalitions under Two-Tier Majority on Regular Graphs
AI总结：
本文研究带自环正则图上基于邻居满意度的两层多数投票机制，建立了图类 $\mathcal{G}_{n|d|h}$ 中存在接受或拒绝提案的标定图的充分必要条件，并探讨其与多数支配等现有文献的联系及性质。
摘要：
本文研究以下问题：考虑一个提案被提交给给定网络 $G$ 的顶点，顶点必须进行投票决定是否接受该提案。每个顶点 $v$ 对提案有自己的估值；若估值为正，则称 $v$ 为“满意”（即期望从采纳提案中获益），若为负则称“不满意”。然而，顶点不仅根据自身估值投票：若顶点 $v$ 的多数邻居对提案满意，则 $v$ 是提案的支持者，反之为反对者。投票结束时，当多数顶点为支持者时，网络集体接受提案。我们研究带自环的正则图上的此问题。具体考虑类 $\mathcal{G}_{n|d|h}$，即阶数 $n$ 为奇数、含全部 $n$ 个自环且有 $h$ 个满意顶点的 $d$-正则图。我们旨在建立 $\mathcal{G}_{n|d|h}$ 包含接受提案的标定图的充分必要条件，以及包含拒绝提案的图的条件。我们还讨论与现有文献（包括多数支配相关研究）的联系，并研究所得条件的性质。
英文摘要：
In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network $G$, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex $v$ has its own valuation of the proposal; we say that $v$ is ``happy'' if its valuation is positive (i.e., it expects to gain from adopting the proposal) and ``sad'' if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex $v$ is a \emph{proponent} of the proposal if the majority of its neighbors are happy with it and an \emph{opponent} in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class $\mathcal{G}_{n|d|h}$ of $d$-regular graphs of odd order $n$ with all $n$ loops and $h$ happy vertices. We are interested in establishing necessary and sufficient conditions for the class $\mathcal{G}_{n|d|h}$ to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
