Graphs with a 3-Cycle-2-Cover

If a graph $$G$$G has three even subgraphs $$C_1$$C1, $$C_2$$C2 and $$C_3$$C3 such that every edge of $$G$$G lies in exactly two members of $$\{C_1, C_2, C_3\}$${C1,C2,C3}, then we say that $$G$$G has a 3-cycle-2-cover. Let $$S_3$$S3 denote the family of graphs that admit a 3-cycle-2-cover, and let $$\mathcal {S}(h,k) = \{G:$$S(h,k)={G:$$G$$G is at most $$h$$h edges short of being $$k$$k-edge-connected$$\}$$}. Catlin (J Gr Theory 13:465–483, 1989) introduced a reduction method such that a graph $$G \in S_3$$G∈S3 if its reduction is in $$S_3$$S3; and proved that a graph in the graph family $$\mathcal {S}(5,4)$$S(5,4) is either in $$S_3$$S3 or its reduction is in a forbidden collection consisting of only one graph. In this paper, we introduce a weak reduction for $$S_3$$S3 such that a graph $$G \in S_3$$G∈S3 if its weak reduction is in $$S_3$$S3, and identify several graph families, including $$\mathcal {S}(h,4)$$S(h,4) for an integer $$h \ge 0$$h≥0, with the property that any graph in these families is either in $$S_3$$S3, or its weak reduction falls into a finite collection of forbidden graphs.