A cyclic edge cut of a graph is an edge cut such that at least two of the resulting connected components each contain at least one cycle. In a connected graph, each smallest cyclic edge cut splits the graph into exactly two components. In a disconnected graph with at least two components that each contain at least one cycle, the empty set of edges comprise a trivial cyclic edge cut.
The cyclic edge cuts of two distinct types for the Petersen graph are illustrated above.
A cyclic edge cut does not exist for all graphs. For example, a graph containing fewer than two cycles cannot have two components each of which contain a cycle. Examples
of graphs having no cyclic edge cuts include the wheel
graphs 
(Dvorák et al. 2004). A graph for which no cyclic edge cut exists may
be taken to have cyclic edge connectivity 
(Lou et al. 2001).
The smallest graph that possesses a cyclic edge cut must contain two triangles (and therefore 6 vertices) connected by a single edge (therefore having a total of 7 edges). The graph so constructed is precisely the 3-barbell graph. There are two next smallest graphs possessing a cyclic edge cut, each having 6 vertices and 8 edges. These graphs are illustrated above.
