A minimum feedback node set in a graph is a minimum node subset whose deletion makes the graph acyclic. Its detection in a dependency graph is important to recover from a deadlock configuration. A livelock configuration is also avoidable if a check point is set in each node in the minimum feedback node set. Hence, its detection is very important to establish dependable network systems. In this letter, we give a minimum feedback node set in a trivalent Cayley graph. Assuming that each word has n bits, for any node, we can judge if it is included in the set or not in constant time.