  Mermin square scenario provides a simple proof for state-independent
contextuality. In this paper, we study polytopes $\text{MP}_\beta$ obtained
from the Mermin scenario, parametrized by a function $\beta$ on the set of
contexts. Up to combinatorial isomorphism, there are two types of polytopes
$\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $\beta$. Our main
result is the classification of the vertices of these two polytopes. In
addition, we describe the graph associated with the polytopes. All the vertices
of $\text{MP}_0$ turn out to be deterministic. This result provides a new
topological proof of a celebrated result of Fine characterizing noncontextual
distributions on the CHSH scenario. $\text{MP}_1$ can be seen as a nonlocal toy
version of $\Lambda$-polytopes, a class of polytopes introduced for the
simulation of universal quantum computation. In the $2$-qubit case, we provide
a decomposition of the $\Lambda$-polytope using $\text{MP}_1$, whose vertices
are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario,
whose vertices are well-known.
