Is there a more interesting name for this graph invariant? It seems to have been called 'complexity' here http://arxiv.org/abs/math/0502579 and here http://www.mathunion.org/ICM/ICM1994.2/Main/icm1994.2.1375.1383.ocr.pdf .
The motivation is that we want to talk about a quantity that is preserved under the graph transformation of collapsing two distinct vertices connected by an edge to a single vertex (thereby removing one edge and one vertex, preserving 'edges minus vertices'). So for example if the quantity 'edges minus vertices plus one' is more natural for some reason and has a name, then this would also be helpful. The concept should not be restricted to e.g. planar graphs.