Orderly generation of simple graphs

In 1978, R. C. Read [246] and I. A. Feuadzhev [72,73] independently published a method suitable for the construction of complete lists of unlabeled discrete structures by evaluating a canonic transversal of the orbits of a finite group G on a finite and totally ordered set (X, ) (which means that for x,x e X either x x or x x is true). For example, the minimal representatives of the orbits form the following canonic transversal of G X, 

we give a general formulation of Read s method to construct this particular transversal. The idea is to start from a suitable set partition 0, A = X of X, consisting of suitable unions il of orbits, to evaluate the elements of the canonic transversal that are contained in Qg. In the next step, the elements of the canonic transversal that belong to are obtained, then the elements of the canonic transversal in so 

1 Algorithm (Orderly generation) We assume an action X of a finite group G on a finite and totally ordered set X. In order to construct the canonic transversal rep (G X) we can proceed as follows  

Choose a suitable set partition of X into nonempty subsets Dj, i e k, which are 

Then we obtain the desired canonic transversal as follows  

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Fig. 5.3. Backtrack tree for orderly generation of simple graphs on three nodes.

Fig. 5.3 Backtrack tree for orderly generation of simple graphs on three nodes.-----------------171...

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