  Schrijver identified a family of vertex critical subgraphs of the Kneser
graphs called the stable Kneser graphs $SG_{n,k}$. Bj\"{o}rner and de
Longueville proved that the neighborhood complex of the stable Kneser graph
$SG_{n,k}$ is homotopy equivalent to a $k-$sphere.
  In this article, we prove that the homotopy type of the neighborhood complex
of the Kneser graph $KG_{2,k}$ is a wedge of $(k+4)(k+1)+1$ spheres of
dimension $k$. We construct a maximal subgraph $S_{2,k}$ of $KG_{2,k}$, whose
neighborhood complex is homotopy equivalent to the neighborhood complex of
$SG_{2,k}$. Further, we prove that the neighborhood complex of $S_{2,k}$
deformation retracts onto the neighborhood complex of $SG_{2,k}$.
