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 a 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 a majority of its vertices are proponents. We study this problem on 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.

翻译：本文研究以下问题：考虑一个场景，提案被提交至给定网络$G$的顶点，顶点必须进行投票并决定接受或否决该提案。每个顶点$v$对提案有自己的估值；若估值为正（即期望从采纳提案中获益），则称该顶点“满意”；若估值为负，则称其“不满意”。但顶点并非仅依据自身估值投票。相反，若顶点$v$的多数邻居对提案满意，则$v$成为提案的*支持者*，反之则为*反对者*。投票结束时，若网络中的多数顶点为支持者，则网络集体接受提案。我们在带环正则图上研究该问题。具体而言，考虑类${\mathcal G}_{n|d|h}$，即阶数$n$为奇数、具有$n$个环和$h$个满意顶点的$d$-正则图。我们旨在建立${\mathcal G}_{n|d|h}$类包含接受提案的标号图的充要条件，以及包含否决提案的标号图的条件。本文还讨论了与现有文献（包括多数支配问题）的联系，并研究了所得条件的性质。