Here - Combinatorial algorithms and algorithmic graph theory

6 Introduction
1.3 Isomorphism-free generation
When generating combinatorial structures, it is important that no isomorphic
copies are output by the generation algorithm. In the literature various techniques
exist to avoid the generation of isomorphic copies. The most commonly
used ones are McKay’s canonical construction path method [89] and the method
of generating canonical representatives (which was independently pioneered by
Faradžev [50] and Read [105]). A good survey of isomorphism rejection techniques
can be found in [14].
Both the method of generating canonical representatives and the canonical
construction path method use a canonical labelling.
If L is the set of all labelled graphs, then a canonical representative function is
a function c : L → L with the properties that for every G ∈ L, c(G) is isomorphic
to G and that c(G) = c(G ′ ) if and only if G and G ′ are isomorphic. The graph
c(G) is also called the canonical representative of G. An isomorphism φ from G
to c(G) is called a canonical labelling of G.
We can implement our own canonical representative function or we can use a
program such as nauty [88] to compute a canonically labelled isomorph of a given
graph. A labelling can for example be represented by a string consisting of the
adjacency matrix of the graph written into one row. The adjacency matrix of a
graph with n vertices is an n × n matrix where the entry of row i and column j
is 1 if and only if there is an edge between vertex i and vertex j, else the entry
is 0. One could then define the labelling of the canonical representative of G to
be e.g. the one with the lexicographically minimal representation.
A straightforward isomorphism rejection technique is isomorphism rejection
by lists. In this method, a list of non-isomorphic graphs which were generated
so far is stored (preferably in the memory for efficiency reasons). Each time a
new graph is generated, it is compared to the graphs from the list. If it is not
isomorphic to any of the graphs in the list, the graph is added to the list, else it is
rejected. Usually the canonical representative is stored rather than the generated
graph, since then one does not have to recompute the canonical representative
of the stored graphs. When the canonical representative of the new graph is
computed, one only has to compare it to the stored canonical representatives.
This approach is of course only feasible if the number of non-isomorphic
graphs is not too big. We use this technique for the generation of irreducible
IPR fullerenes (see Section 4.3) and minimal Ramsey graphs (see Section 5.3).
The method of generating canonical representatives only accepts graphs if
their labelling is canonical. The difficulty of this method is to define a canonical