We consider the stability of robust scheduling policies for Lu-Kumar networks, i.e., open networks with arbitrary routing matrix and several disjoint groups of queues in which at most one queue can be served at a time in each group. A scheduling policy is called robust if it does not depend on the arrival and service rates nor on the routing probabilities. We propose two robust polices: longest-queue scheduling and a new policy called longest-dominating-queue scheduling. We show that longest-queue scheduling is throughput-optimal for two groups of two queues. We also prove the throughput optimality of longest-dominating-queue scheduling when the network topology is acyclic. In the end, we maximize utility of these networks using a robust distributed scheme based on local book-keeping.