  Given a graph $G$, the $G$-parking function ideal $M_G$ is an artinian
monomial ideal in the polynomial ring $S$ with the property that a linear basis
for $S/M_G$ is provided by the set of $G$-parking functions. It follows that
the dimension of $S/M_G$ is given by the number of spanning trees of $G$, which
by the Matrix Tree Theorem is equal to the determinant of the reduced Laplacian
of $G$. The ideals $M_G$ and related algebras were introduced by Postnikov and
Shapiro where they studied their Hilbert functions and homological properties.
The author and Sanyal showed that a minimal resolution of $M_G$ can be
constructed from the graphical hyperplane arrangement associated to $G$,
providing a combinatorial interpretation of the Betti numbers.
  Motivated by constructions in the theory of chip-firing on graphs, we study
certain `skeleton' ideals $M_G^{(k)} \subset M_G$ generated by subsets of
vertices of $G$ of size at most $k+1$. Here we focus our attention on the case
$k=1$, the $1$-skeleton of the $G$-parking functions ideals. We consider
standard monomials of $M_G^{(1)}$ and provide a combinatorial interpretation
for the dimension of $S/M_G^{(1)}$ in terms of the signless Laplacian for the
case $G = K_{n+1}$ is the complete graph. Our main study concerns homological
properties of these ideals. We study resolutions of $M_G^{(1)}$ and show that
for a certain class of graphs minimal resolution is supported on decompositions
of Euclidean space coming from the theory of tropical hyperplane arrangements.
This leads to combinatorial interpretations of the Betti numbers of these
ideals.
