Basic Trope: A character tries to get his/her significant other to dump him/her first by acting like a Jerkass and/or being disgusting. It rarely works.
Straight: Alice and Bob are together, but Bob has had enough of the relationship for whatever reason(s). Bob tries to get Alice to dump him first by doing things like flirting with another woman in front of her, eating a Philly cheesesteak sandwich (with extra onion!) before kissing her, cussing more than usual, and knocking her favorite soap opera. For her part, Alice doesn't see this as a big deal.
Downplayed: Bob has been finding his relationship with Alice less satisfying lately, and begins being less attentive as other things occupy his interest.
Bob doesn't want to be the bad guy and hurt Alice's feelings. He thinks she'll be less hurt if she does the dumping.
Alice and Bob have a prenup and/or live in a place with certain divorce laws that make it unfavorable for Bob to be the one to initiate divorce.
The reason Bob wants to break up with Alice is something petty and shallow, such as Alice not being into the whole Brazilian wax (or shaving "down there") thing. Bob doesn't want to be The Guy Who Dumped His Girlfriend Over Her Pubic Hair, but he can't think of any other reason to dump her, so he figures he'll give her a reason to dump him so he won't have to come up with a more serious reason to dump her.
Bob wants to break up with Alice, so he starts acting extra nice to her.
Bob's relationship with Alice is on the rocks, so he starts acting extra nice to her in hopes that she'll reconsider breaking up with him.
Bob is interested in Alice, but she is Oblivious to Love, so he starts acting like a jerk towards her to get her to notice him.
Gender Inverted: Alice and Bob are together, but Alice wants to call it quits. So she starts acting like The Lad-ette (rude and disgusting) in hopes that Bob will dump her. Instead of dumping her, he finds her sweatpants, beer-chugging, cussing, unabashed farting, and unshaven legs more attractive.
Bob starts acting like a disgusting Jerkass, but then feels bad about treating Alice like that.
Invoked: Bob loves his girlfriend Alice, but finds himself perturbed by the fact that she doesn't shave, wax, or even trim "down there." Aside from that, he loves everything else about her, and genuinely cares about her feelings.
Exploited: Bob and Alice are in an arranged marriage that Bo wants to get out of, but can't for whatever reason, so he tries to make Alice break it off instead.
Bob decides he's not going to let something so petty and shallow bother him or jeopardize their relationship. (He might find a way to get the message across without being hurtful, such as shaving her down during sexytime and gushing over how good it looks.)
Or if the relationship is troubled by more than just Alice's ladybush, he decides to just be honest and gentle with her.
Discussed: "Just get rid of her the way I got rid of my last girlfriend: by being a jerk!"
Deconstructed: Not the best way of going about it; all it does is make Bob look like a Jerkass...and a Dirty Coward to boot. (And that's if Alice doesn't fall victim to All Girls Want Bad Boys and find his disrespectful behavior endearing or charming.)
Reconstructed: Bob decides that a better way to go about it is to simply tell Alice how he feels about the relationship, and let her down as gently as possible.