  Zonotopal algebras (external, central, and internal) of an undirected graph G
introduced by Postnikov-Shapiro and Holtz-Ron, are finite-dimensional
commutative graded algebras whose Hilbert series contain a wealth of
combinatorial information about G. In this paper, we associate to G a new
family of algebras, which we call bizonotopal, because their definition
involves doubling the set of edges of G. These algebras are monomial and have
intricate properties related, among other things, to the combinatorics of
graphical parking functions and their polytopes.
  Unlike the case of usual zonotopal algebras, the Hilbert series of
bizonotopal algebras are not specializations of the Tutte polynomial of G.
Still, we show that in the external and central cases these Hilbert series
satisfy a modified deletion-contraction relation. In addition, we prove that
the external bizonotopal algebra is a complete graph invariant.
