Assume we have bridgeless cubic graph $G(V, E)$, $n=|V|$. By Petersen's theorem, every such graph has a perfect matching. Moreover, given any edge in $G$ there exists a perfect matching containing this edge. I am interested in finding the size $|C|$ of the smallest set of vertex-disjoint edges $C$ that uniquely determines a perfect matching $M$ (i.e. $C \subset M$).

Notice that every cubic bridgeless graph G has at least $ \Omega (2^{|V(G)|/3656})$ perfect matchings.

What is the minimum number of vertex-disjoint edges $|C|$ that uniquely force any perfect matching in any bridgeless cubic graph?

Equivalently, a subset $C$ of a perfect matching $M$ is called a forcing set if $M$ is the unique perfect matching containing $C$. The forcing number of $M$ is the minimum cardinality of forcing sets of $M$. The forcing number of a cubic bridgeless graph $G(V, E)$ is the minimum number of forcing numbers of all perfect matchings of $G$. So my questions becomes:

What is the forcing number for any bridgeless cubic graph?

I guess a constant is not enough. I guess that we need at least $|C|=\Omega( n)$ and $O(\log n)$ is not enough. Is there a proof?