  We define two classes of topological infinite degree covering maps modeled on
two families of transcendental holomorphic maps. The first, which we call
exponential maps of type $(p,q)$, are branched covers and is modeled on
transcendental entire maps of the form $P e^{Q}$, where $P$ and $Q$ are
polynomials of degrees $p$ and $q$. The second is the class of universal
covering maps from the plane to the sphere with two removed points modeled on
transcendental meromorphic maps with two asymptotic values. The problem we
address is to give a combinatorial characterization of the holomorphic maps
contained in these classes whose post-singular sets are finite. The main
results in this paper are that a post-singularly finite topological exponential
map of type $(0,1)$ or a certain post-singularly finite topological exponential
map of type $(p,1)$ or a post-singularly finite universal covering map from the
plane to the sphere with two points removed is combinatorially equivalent to a
holomorphic same type map if and only if this map has bounded geometry.
