  We investigate Hadwiger's conjecture for graphs with no stable set of size 3.
Such a graph on at least 2t-1 vertices is not t-1 colorable, so is conjectured
to have a $K_t$ minor. There is a strengthening of Hadwiger's conjecture in
this case, which states that there is always a minor in which the preimages of
the vertices of $K_t$ are connected subgraphs of size one or two. We prove this
strengthened version for graphs whose complement has an even number of vertices
and fractional chromatic number less than 3. We investigate several possible
generalizations and obtain counterexamples for some and improved results from
others. We also show that for sufficiently large $n=|V(G)|$, a graph with no
stable set of size 3 has a $K_{1/9 n^{4/5}}$ minor using only sets of size one
or two as preimages of vertices.
