Inu Yasha is a really good anime series. *ahem* anywayz,
The word relationship is used to mean a number of different things. It can refer to the interactions between two people (I will use Jack and Jill). It can refer to said interactions, and the emergent properties of those interactions. It can refer to only the emergent properties. It can refer to an actual thing, that supposedly exists, and has consequences (I hold this view is false). If I say "relationships aren't things" or "relationships don't exist" that's what I'm referring to, though I try to be more clear than that.
Or sometimes people say "you should stick together, for the sake of the relationship." In this case, relationship is shorthand for the valuable knowledge of each other, convergence, incomplete joint-projects, and such that the people have.
Reductionist relationship is a good term for just the interactions. This would include physical specifications on body positions for the time Jack and Jill went to ... not "the pizza parlor" but some set of lattitude and longitude coordinates. And for everywhere else they had gone they met some specifications about proximity or sounds directed at each other or something. It would include what sounds they made, but not what the words meant.
Emergent relationship is a good term for talking about emergent properties of the reductionist relationship, without bringing up anything of the information in the reductionist description. This would include how Jack and Jill feel about each other, what they mean to each other, and Jack's obligation to show up at Jill's house at 8pm on Tuesday (because he said he would).
I consider "relationship" to mean both of these. Anyway, you will notice that all the emergent properties are direct results of various interactions between Jack and Jill. The term "relationship" simply refers to multiple things at once. It is not itself a thing, with properties. Why does this matter?
Some people claim that relationships bring about obligations or various other consequences, in and of themselves. Example obligations are: to stay together, to not fuck other people, to be nice, to be supportive, to not leave abruptly, or to take care of one's partner in times of need. This is false and harmful. (Or, one could make the case it's misleading, harmful, semi-true shorthand). [Some or all of the things mentioned may be implied by the morality of the situation in some relationships]
The reductionist view of relationships as various interactions and their emergent properties is valid, and I think useful for seeing certain things, but for many things is a bad idea. It makes a lot of calculations (like predicting whether there will be a breakup in the next 2 years) totally infeasible, and it can often obscure the morality of a situation. So, while I often use it to answer theory questions, it's not that useful for many real-life things.