Here - Combinatorial algorithms and algorithmic graph theory

26 Generation of cubic graphs
Figure 2.8: A prime graph which contains 2 lollipops.
• G contains 1 reducible lollipop and no reducible non-adjacent edge diamonds.
Here there are exactly 3 edges which can be chosen for expansion: the 3
edges which are incident to the central vertex of the reducible lollipop. If
the expansion is applied to other edges of the lollipop, the expanded graph
will not be prime.
• G contains reducible 1 non-adjacent edge diamond and no reducible lollipops.
The only way to destroy the reducible non-adjacent edge diamond without
generating a new one (and where the expanded graph is still prime) is to
apply the expansion to the edges where one vertex is an extremal vertex of
the K4
− of the reducible non-adjacent edge diamond and where the other
vertex of the edge is the neighbour of that extremal vertex which is not part
of the K4 − . So there are exactly 2 edges which can be chosen for expansion.
• G contains no reducible lollipops and no reducible non-adjacent edge diamonds.
In this case all edges which are no diamond edges are eligible for expansion.
• In any other case the list of edges which are eligible for expansion is empty:
since the edge diamond insertion operation only modifies one edge of the
original graph, it cannot destroy more than 1 reducible lollipop or reducible
non-adjacent edge diamond (except for the graph from Figure 2.8).
The edge diamond insertion operation only modifies one edge in the original
graph and the new edges of the expanded graph are irreducible since they share a
vertex with a K4 − . Since we do not apply this operation to diamond edges (unless
the parent graph is K 4 ), the expanded graph will be prime if the original graph
was. So for the edge diamond operation we do not have to check if the expanded
graphs are prime.