The role of stray dogs in the persistence of domestic dog rabies, and whether removal of such dogs is beneficial, remains contentious issues for control programs seeking to eliminate rabies. While a community might reach the WHO vaccination target of 70% for dogs that can be handled, the stray or neighborhood dogs that are too wary of humans to be held are a more problematic population to vaccinate. Here, we present a method to estimate vaccination targets for stray dogs when the dog population is made up of stray, free-roaming, and confined dogs, where the latter two types are considered to have an identifiable owner. The control effort required for stray dogs is determined by the type-reproduction number, T1, the number of stray dogs infected by one rabid stray dog either directly or via any chain of infection involving owned dogs. Like the basic reproduction number R0 for single host populations, T1 determines the vaccination effort required to control the spread of disease when control is targeted at one host type, and there is a mix of host types. The application of T1 to rabies in mixed populations of stray and owned dogs is novel. We show that the outcome is sensitive to the vaccination coverage in the owned dog population, such that if vaccination rates of owned dogs were too low then no control effort targeting stray dogs is able to control or eliminate rabies. The required vaccination level also depends on the composition of the dog population, where a high proportion of either stray or free-roaming dogs implies unrealistically high vaccination levels are required to prevent rabies. We find that the required control effort is less sensitive to continuous culling that increases the death rate of stray dogs than to changes in the carrying capacity of the stray dog population.