Regular tessellations of the hyperbolic plane by fundamental domains of a Fuchsian group

For positive integers $p$ and $q$ with $1/p+1/q<1/2$, a tessellation of type $\{p,q\}$ is a tessellation of the hyperbolic plane by regular $p$-gons with $q$ $p$-gons meeting at each vertex. In this paper, a necessary and sufficient condition on the integers $p$ and $q$ is established to determine when a tessellation of type $\{p,q\}$ can be realized as a tessellation of the hyperbolic plane by fundamental domains of some Fuchsian group. Specifically, a tessellation of type $\{p,q\}$ is a tessellation by fundamental domains if and only if $q$ has a prime divisor less than or equal to $p$.