We characterize the monotone bounded depth formula complexity for graph homomorphism and colored isomorphism polynomials using a graph parameter called the cost of bounded product depth baggy elimination tree. Using this characterization, we show an almost optimal separation between monotone circuits and monotone formulas using constant-degree polynomials for all fixed product depths, and an almost optimal separation between monotone formulas of product depths $Δ$ and $Δ$ + 1 for all $Δ$ $\ge$ 1.

翻译：我们利用称为有界乘积深度宽松消去树代价的图参数，刻画了图同态与着色同构多项式的单调有界深度公式复杂性。基于该刻画，我们证明了对于所有固定乘积深度，利用常数度多项式可实现单调电路与单调公式的近乎最优分离；同时对于所有Δ≥1，实现了乘积深度Δ与Δ+1的单调公式间的近乎最优分离。