  We extend the theory of combinatorial link Floer homology to a class of
oriented spatial graphs called transverse spatial graphs. To do this, we define
the notion of a grid diagram representing a transverse spatial graph, which we
call a graph grid diagram. We prove that two graph grid diagrams representing
the same transverse spatial graph are related by a sequence of graph grid
moves, generalizing the work of Cromwell for links. For a graph grid diagram
representing a transverse spatial graph $f:G \rightarrow S^3$, we define a
relatively bigraded chain complex (which is a module over a multivariable
polynomial ring) and show that its homology is preserved under the graph grid
moves; hence it is an invariant of the transverse spatial graph. In fact, we
define both a minus and hat version. Taking the graded Euler characteristic of
the homology of the hat version gives an Alexander type polynomial for the
transverse spatial graph. Specifically, for each transverse spatial graph $f$,
we define a balanced sutured manifold $(S^3\smallsetminus f(G), \gamma(f))$. We
show that the graded Euler characteristic is the same as the torsion of
$(S^3\smallsetminus f(G), \gamma(f))$ defined by S. Friedl, A. Juh\'{a}sz, and
J. Rasmussen.
  This paper is dedicated to the memory of Tim Cochran.
