The Fast Probabilistic Consensus on a Set (FPCS) is a leaderless voting consensus protocol designed for achieving agreement among nodes on a preferred maximal independent set within a graph of conflicts. The protocol's robustness and efficiency have been previously established for complete graphs under the security threshold of q < β < 1/3, where q represents the proportion of Byzantine nodes. In this paper, we analyze a protocol edge case -a star graph. We show that the security threshold found for complete graphs is not restrictive enough and conjecture a new threshold of q < β < 1/4. This advance highlights the tradeoff between versatility and reduced security, showing the protocol's adaptability across a broader range of scenarios at the cost of tighter security constraints.