Two answers.I'm thinking about trees because of the two out of three property:
For a simple (no multiple edges) undirected graph G, any 2 of the three conditions

 1. cycle-free (better **acyclic**) 
 2. connected
 3. #edges=#vertices-1

Means G is a tree. 

Of course that isn't what you asked. Condition three is not one word although we could coin **uni-deficicient**

so **simple+undirected+acyclic+connected** defines tree.


Certain sets are relations (basically any set of ordered pairs). Not counting that as a condition

For a relation, **reflexive+symmetric+transitive** defines **Equivalence relation**

similarly **reflexive+antisymmetric+transitive** defines **partial order**

Actually there are $k$-ary relations for other $k$ so one could  quantify over all relations and restrict to **binary** relations.