  Let $G$ be a graph with $n$ vertices and $m$ edges. One of several
hierarchies towards the stability number of $G$ is the exact subgraph hierarchy
(ESH). On the first level it computes the Lov\'{a}sz theta function
$\vartheta(G)$ as semidefinite program (SDP) with a matrix variable of order
$n+1$ and $n+m+1$ constraints. On the $k$-th level it adds all exact subgraph
constraints (ESC) for subgraphs of order $k$ to the SDP. An ESC ensures that
the submatrix of the matrix variable corresponding to the subgraph is in the
correct polytope. By including only some ESCs into the SDP the ESH can be
exploited computationally.
  In this paper we introduce a variant of the ESH that computes $\vartheta(G)$
through an SDP with a matrix variable of order $n$ and $m+1$ constraints. We
show that it makes sense to include the ESCs into this SDP and introduce the
compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems
favorable as the SDP is smaller. However, we prove that the bounds based on the
ESH are always at least as good as those of the CESH. In computational
experiments sometimes they are significantly better.
  We also introduce scaled ESCs (SESCs), which are a more natural way to
include exactness constraints into the smaller SDP and we prove that including
an SESC is equivalent to including an ESC for every subgraph.
