  We introduce a notion of graph homeomorphisms which uses the concept of
dimension and homotopy for graphs. It preserves the dimension of a subbasis,
cohomology and Euler characteristic. Connectivity and homotopy look as in
classical topology. The Brouwer-Lefshetz fixed point leads to the following
discretiszation of the Kakutani fixed point theorem: any graph homeomorphism T
with nonzero Lefschetz number has a nontrivial invariant open set which is
fixed by T.
