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Enumeration and generation of a class of regular digraphs. (English) Zbl 0652.05020
Let $$G=(V,A)$$ be a finite digraph; and $$X\subseteq A$$ is called an alternating cycle (ac) iff its arcs can be ordered as $$e_ 0,e_ 1,...,e_{2r-1}$$ such that $$e_ i$$ and $$e_{i\oplus 1}$$ $$(i=0,1,2,...,2r-1$$ and $$i\oplus 1$$ denotes addition mod 2r) have a common end-vertex which is called exit-vertex of X (start-vertex which is called entry-vertex of X) if i is even (odd), whereby loops are allowed but not parallel arcs. In the present article is studied the class of those regular digraphs, whose vertices all have indegree $$=$$ outdegree $$=2$$, these are the so-called 2-diregular digraphs (2-dds). It is known that a 2-dd can be uniquely decomposed into acs in linear time, and by using this property it is developed an interesting enumeration formulae for 2-dds and an algorithm to generate them efficiently.
Let be $$A=\{X_ 1,X_ 2,...,X_ k\}^ a$$collection of vertex- disjoint simple acs of G. It is introduced a bijection from the set of entry-vertices of some $$X_ i$$ into set of exit-vertices of some $$X_ i$$ and in the set of all these bijections is defined the following equivalence relation: two bijections are equivalent iff they yield isomorphic 2-dds, and a partial order. In the article is shown that it suffices to consider the simple acs into which any 2-dd can be split and the given bijections. By using many known results the authors get the number of nonisomorphic 2-dds and from this the number of nonisomorphic connected 2-dds. The computations for these with up to 20 vertices were done on a computer.
In a last chapter the generation procedure which uses a backtracking procedure for generating permutations in lexicographic order is described in 3 steps. Also by using a computer programming 2-dds have been considered with up to 11 vertices.
Reviewer: H.-J.Presia

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C30 Enumeration in graph theory
##### Keywords:
2-diregular digraphs; 2-dds; enumeration; generation
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##### References:
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