On the hardness of turn-angle-restricted rectilinear cycle cover problems

A cycle cover of a graph G is a collection of disjoint cycles that spans G. Generally, a (possibly disconnected) cycle cover is easier to construct than a connected (Hamiltonian) cycle cover. One might expect this since the cycle cover property is local whereas connectivity is a global constraint. We compare the hardness of CONNECTED CYCLE COVER and CYCLE COVER under various constraints (both local and global) on the orientation, crossings, and turning angles of edges. Surprisingly perhaps, under specific constraints, the cycle cover problem is NP-hard whereas the corresponding connected cycle cover problem can be solved in polynomial time.