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.

翻译：$G 是一个以美元为顶端和美元+美元+1美元限制的图表。 在美元水平上,它增加了所有精确的分层限制(ESC), 用于排序的分层值为$G美元。 精确的分层等级(ESH) 之一就是精确的分层等级。 在第一个层次上,它计算Lov\\\ {a}sz sta 函数Lovtheta(G) 为半确定性(SDP), 矩阵变量为n+1美元和n+m+1美元。 在美元水平上, 它增加了所有精确的分层限制(ESC) 用于排序的分层限制(ESC ) 美元到 SDP 的分层值( ESCF) 。 在一个比例上, ESCH 对应的矩阵变量的子矩阵值值在正确的分级值上, 将 ESCH 的子矩阵值值(ESH ) 更小的子值(ESCE) 也显示将ESC 的精度纳入到SD 的精度。