  Let $M$ be a compact orientable irreducible 3-manifold and $H$ be an
unstabilized genus three Heegaard splitting of $M$. In this article, we will
define a simplicial complex of weak reducing pairs for $H$ and find several
properties of this complex. Using this method, we will prove that an
unstabilized Heegaard splitting of genus three is critical in a certain
condition. In addition, we will show that the standard genus three Heegaard
splitting for $T^3$ and the induced Heegaard splitting of the three component
chain exterior by a certain tunnel system are critical as examples of the main
theorem.
