they are having a domestic (informal)
they are having an argument
they have blazing rows
they have very serious, often noisy arguments
(s)he is playing away from home
(s)he is having an affair
to break up with somebody
to finish a relationship with somebody
to split up
to break up
to dump somebody
to finish a relationship with somebody without much thought for the other person
break-up lines
(fixed) expressions, often sweet or apologetic, used to tell a partner, girlfriend or boyfriend that the relationship is finished or is going to finish: