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 $SG_{ 2,k}$.
Categories: math.CO