  In this paper we consider a natural extremal graph theoretic problem of
topological sort, concerning the minimization of the (topological)
connectedness of the independence complex of graphs in terms of its dimension.
We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the
connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of
bipartite graphs $G$ is tight. In our main theorem we characterize the extremal
examples. Our proof of this characterization is based on topological machinery.
  Our motivation for studying this problem comes from a classical conjecture of
Ryser. Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph
has a vertex cover of size at most $(r - 1)$-times the size of the largest
matching. For $r = 2$, the conjecture is simply K\"onig's Theorem. It has also
been proven for $r = 3$ by Aharoni using a beautiful topological argument. In a
separate paper we characterize the extremal examples for the $3$-uniform case
of Ryser's Conjecture (i.e., Aharoni's Theorem), and in particular resolve an
old conjecture of Lov\'asz for the case of Ryser-extremal $3$-graphs.
  Our main result in this paper will provide us with valuable structural
information for that characterization. Its proof is based on the observation
that link graphs of Ryser-extremal $3$-uniform hypergraphs are exactly the
bipartite graphs we study here.
