# Vertex Covers whose vertex induced subgraph has an even number of edges and no isolated vertices

Let $G$ be a graph, and let $C_{E,0}$ be the number of those vertex covers of $G$ satisfying both the following properties:

• Their corresponding vertex induced subgraph has an even number of edges.
• Their corresponding vertex induced subgraph has no isolated vertices.

Questions

1. Did anyone already study the problem of computing $C_{E,0}$? What is known about it?
2. More importantly, which is the computational complexity of computing $C_{E,0}\ mod\ 2$?