The cyclic and dihedral groups can be made to act on the set Acyc(Y) of
acyclic orientations of an undirected graph Y, and this gives rise to the
equivalence relations ~kappa and ~delta, respectively. These two actions and
their corresponding equivalence classes are closely related to combinatorial
problems arising in the context of Coxeter groups, sequential dynamical
systems, the chip-firing game, and representations of quivers.
In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y)
and whose connected components encode the equivalence classes. The number of
connected components of these graphs are denoted kappa(Y) and delta(Y),
respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y)
can be derived from kappa(Y), and give enumeration results for kappa(Y).
Moreover, we show how to associate a poset structure to each kappa-equivalence
class, and we characterize these posets. This allows us to create a bijection
from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y'
and Y'' denote edge deletion and edge contraction for a cycle-edge in Y,
respectively, which in turn shows that kappa(Y) may be obtained by an
evaluation of the Tutte polynomial at (1,0).Comment: The original paper was extended, reorganized, and split into two
papers (see also arXiv:0802.4412
