Graphs with a 3-Cycle-2-Cover

Abstract

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