spatial graphs which had been introduced by Wu [@Wu; @1960; @Wu; @1965], and showed that the Wu invariant can be defined combinatorially from a graph projection. However, the Wu invariant is not always easy to compute, and (like the Simon invariant) depends on the choice of labeling of the vertices of a graph. For this reason, the role of the Wu invariant in distinguishing a spatial graph from its mirror image has been limited to showing that for any embedded non-planar graph $\Gamma$, there is no orientation reversing homeomorphism of $(S^3,\Gamma)$ that fixes every vertex of $\Gamma$ (see [@Nikkuni; @2006]). Without this restriction on the vertices, many non-planar graphs including $K_5$ and $K_{3,3}$ have achiral embeddings as shown in Figure \[achiral\].

![Achiral embeddings of $K_5$ and $K_{3,3}$.[]{data-label="achiral"}](achiral.eps){width="50.00000%"}

In this paper, we define numerical invariants that are obtained by reducing the Wu invariant and by generalizing the Simon invariant. We then use our invariants to prove that no matter how the complete graph $K_7$, the Möbius ladders $M_{2N+1}$, and the Heawood graph are embedded in $S^3$, there is no orientation reversing homeomorphism of $S^3$ which takes the embedded graph to itself. Finally, we show that our invariants can be used to 