You need to understand her nature first. It looks like,she's not completely monogamous type of person! She is not even maintaining serial monogamy type relations! It looks like,your girl friend is a promiscuous type of person and wants to keep relation with multiple mates. She would keep swinging like this between partners,and won't be able to decide which one to finally choose!
Now,if you can't accept her lying,breaking up and coming back again,keeping parallel relations with another male,etc,you need to talk with her about that! Tell her that you won't give her a second chance if she breaks up again or meet the guy again!!
However,if you want to accept her the way she is now,then that is another issue! May be,you could let her think about that guy romantically! May be,you could accept that she would keep relation with both of you. This may seem strange to you,but some people do let their loved one to have multiple relations!!
The point is,you have to take firm decision-whatever that is! Otherwise,you will remain in an unhappy relation,and stay sad in this relation as long as it is there!!
Communicate with her about your feelings,thoughts and wants openly and honestly! See what she has to say and what she actually does! Also,ask yourself what you want,how far you can go in this relation.....And take steps accordingly!!
You have to decide whether you want to have this relation and stay unhappy in the days to come,or you would accept things as they are,and try to be happy accepting everything as they are!