Petersen's theorem states that a bridgeless cubic graph contains a perfect matching (1-factor). Motivated by this question, Complexity of finding 2 vertex-disjoint (|V|/2)-cycles in cubic graphs? I am interested in the problem of deciding the existence of perfect matching $M$ such that removing the edges of $M$ leaves two node-disjoint cycles of equal cardinality (each cycle has size $|V|/2$).
Is this problem solvable in P-time? Is it $NP$-complete?
The input is connected bridgeless cubic graph.