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.

翻译：本文研究以下问题：考虑一个场景，给定网络$G$中的顶点需针对某项提案进行投票，决定接受或拒绝该提案。每个顶点$v$对提案有各自的估值：若估值为正（即预期从采纳提案中获益），则称该顶点为“快乐”；若估值为负，则称其为“悲伤”。然而，顶点并非仅依据自身估值投票。具体而言，若某顶点$v$的多数邻居对提案感到快乐，则$v$成为提案的“支持者”；反之则为“反对者”。投票结束时，当网络中的多数顶点成为支持者时，集体接受该提案。我们针对带自环的正则图研究此问题。具体考虑具有奇数阶$n$、全部带自环且包含$h个快乐顶点的$d$-正则图类$\mathcal{G}_{n|d|h}$。我们旨在建立$\mathcal{G}_{n|d|h}$存在接受提案的标记图的充分必要条件，以及存在拒绝提案的图的条件。同时讨论本研究与现有文献（包括多数支配问题）的联系，并探究所得条件的性质。