Counting/Enumerating Minimal Edge Covers

Question

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?

Answer 1

• Counting edge covers is #P-complete. You can find a proof here Graph Orientations with No Sink and an Approximation for a Hard Case of #SAT. See also the stack-exchange discussion. One can probably modify the proof to show #P-completeness for minimal edge covers.

• enumerating (minimal) edge covers can be done with polynomial delay as the hypergraph has bounded degree (see New Results on Monotone Dualization and Generating Hypergraph Transversals)

• the following Low-Exponential Algorithm for Counting the Number of Edge Covers on Simple Graphs claims an $O(1.465575^{(m−n)} × (m + n))$ time algorithm. I am wondering whether one cannot do better.