I think there are lot of similar problems. Here are two in vertex version and one in edge version:

1) Does a given graph have an **independent** feedback vertex set? (we don't care about the size of the set). 
This problem is NP-complete; the proof can be derived from the proof of Theorem 2.1 in 
[Garey, Johnson & Stockmeyer](http://www.sciencedirect.com/science/article/pii/0304397576900591#).

2) Does a given graph have a vertex cover **that induces a tree**? (we don't care about the size of the set). 
This [paper](http://www.sciencedirect.com/science/article/pii/S0166218X98001164) 
gives an NP-completeness proof for this problem (Theorem 2); even for bipartite graphs.

3) Does a given graph have a dominating edge set **the edges of which form an induced $1$-regular subgraph**? 
(also known as dominating induced matching or efficient edge dominating; the vertex version is given in the 
second answer by Mohammad. 
Again, we don't care about the size of the set). 
This problem is NP-complete (well-known, first proved [here](http://www.sciencedirect.com/science/article/pii/002001909390084M)), even for planar bipartite graphs. 

The first two problems are particular examples of the problem class called stable-$\pi$: 
Let $\pi$ be a graph property. Does a given graph have a vertex cover satisfying $\pi$? 
More NP-complete cases as well as polynomially solvable cases can be found in 
[this](http://www.sciencedirect.com/science/article/pii/S0166218X98001164) 
and in [this](http://www.sciencedirect.com/science/article/pii/S0166218X13003119) paper 
(and the refs given there) .