= Bicyclic kernel classification
The <graph kernel> of a <bicyclic graph> satisfies $\sum_v(\deg v-2)=2$, with all degrees at least three. It therefore has either one degree-four vertex carrying two loops, a <bouquet graph>, or two degree-three vertices. In the latter case connectivity and degree counting give either three parallel edges, the three-edge <Theta graph>, or one edge between the vertices and one loop at each, a <dumbbell graph>. All bicyclic cores are subdivisions of these three kernels, subject to simplicity restrictions on loop lengths and parallel paths.
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