A cyclic edge cut of a connected 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 minimum
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 
comprises a trivial cyclic edge cut (Dvorák et al. 2004).
A cyclic edge cut of smallest possible size in a given graph is called a minimum cyclic edge cut.
The cyclic edge cuts of two distinct types for the Petersen graph 
are illustrated above. Each involves five edges, so the cyclic
edge connectivity is 
.
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.
