Airlines were required to pick up some of each other’s passengers in the event of a carrier shutdown as a result of receiving government aid post 9/11 but there are limits (I don’t happen to know the specifics); most of National's passengers got home on other airlines because of this requirement but it may not apply any longer. Other than that, there is no requirement that airlines must carry each other’s passengers; each airline signs agreements with other airlines in order to carry tickets; some airlines like Southwest do not have ticketing agreements with any other airlines.