This is a follow-up to a previous question. What graphs have incidence matrices of full rank?

Obvious members of the class: complete graphs.

Obvious counterexamples: Graph with more than two vertices but only one edge.

I'm tempted to guess that the answer is graphs that with contain spanning trees as subgraphs. However, I haven't put much thought into this.

This is a follow-up to a previous question. What graphs have incidence matrices of full rank?

Obvious members of the class: complete graphs.

Obvious counterexamples: Graph with more than two vertices but only one edge.

I'm tempted to guess that the answer is graphs that with contain spanning trees as subgraphs. However, I haven't put much thought into this.