Here - Combinatorial algorithms and algorithmic graph theory

22 Generation of cubic graphs
2. G contains no reducible lollipops, but contains at least one reducible nonadjacent
edge diamond.
The proof is completely analogous to the proof of 1.
3. G contains no reducible lollipops and no reducible non-adjacent edge diamonds,
but at least one reducible edge diamond.
The proof is completely analogous to the proof of 1.
Proof of part 2. Assume that all prime graphs with less than n vertices
(n > 4) are generated at most once by the algorithm.
Suppose that the algorithm generated two isomorphic prime graphs G and G ′
with n vertices which are both accepted by the algorithm. The following cases
can occur:
1. G contains at least one reducible lollipop.
In this case G and G ′ were constructed by a lollipop insertion operation. So
we apply the canonical reduction to G and G ′ and call the reduced graphs
p(G) and p(G ′ ) respectively. It follows from Lemma 2.2 that p(G) and
p(G ′ ) are prime. Suppose the lollipop l was reduced in G and the lollipop
l ′ in G ′ and let γ be an isomorphism from G to G ′ . Since l and l ′ are both
canonical reducible lollipops, γ(l) is in the same orbit as l ′ under the action
of Aut(G ′ ), so p(G) ∼ = p(G ′ ). By induction p(G) = p(G ′ ).
Suppose that G was constructed from p(G) by applying the lollipop insertion
operation to the edge e of p(G) and that G ′ was obtained by applying
the lollipop insertion operation to edge e ′ . But restricting the isomorphism
γ (that maps l to l ′ ) to the vertices that already belonged to p(G) gives
an automorphism of p(G) mapping e to e ′ , showing that they were in the
same orbit of Aut(p(G)), contrary to our procedure.
2. G contains no reducible lollipops, but contains at least one reducible nonadjacent
edge diamond.
The proof is completely analogous to the proof of 1.
3. G contains no reducible lollipops and no reducible non-adjacent edge diamonds,
but at least one reducible edge diamond.
The proof is completely analogous to the proof of 1.