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The graph kernel of a bicyclic graph satisfies ${\sum }_v(\mathrm{deg}v-2)=2$, 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.

A bouquet graph consists of one vertex and $j$ self-loops. Its vertex has degree $2j$, and its graph excess is $j-1$. The two-loop bouquet is one of the three possible kernels of a bicyclic graph; in a simple subdivision, each loop must expand to a cycle of length at least three.