The Cycle double cover (CDC) conjecture states that for every bridgeless graph G, there exists a family ℱ of cycles such that each edge of the graph is contained in exactly two members of ℱ. Given an embedding of a graph G, an edge e is called a singular edge if it is visited twice by the boundary of one face. The CDC conjecture is equivalent to bridgeless cubic graphs having an embedding with no singular edge. In this work, we introduce nontrivial upper bounds on the minimum number of singular ...