Grid homology for spatial graphs and a Künneth formula of connected sum

In this paper, we research the grid homology for spatial graphs with cut edges. We show that the grid homology for spatial graph $f$ is trivial if $f$ has sinks, sources, or cut edges. As an application, we give purely combinatorial proofs of some formulas including a Künneth formula for the knot Floer homology of connected sums in the framework of the grid homology.