# Counting/Enumerating Minimal Edge Covers

A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it.

Questions

1. Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial upper bound on their number? Is there any clever algorithm in literature to enumerate them?
2. Empirically, I'm observing that the number of Minimal Edge Covers seems to be always odd: so far in my experiments I've never met a graph having an even number of them. Is this a theoretically proven fact? If not, has their parity been investigated in literature?
• Doesn't $C_4$, the cycle graph on 4 vertices, have two minimal edge covers? – Yonatan N Nov 4 '18 at 18:59