introduce a graph. The *row-column graph* of a $t\times u$ matrix $M$ is the bipartite graph on the vertices $x_1$, $\dots$, $x_t$, $y_1$, $\dots$, $y_u$ where $x_k$ is adjacent to $y_\ell$ if and only if $M_{k,\ell}\neq 0$. Albert, Atkinson, Bouvel, Ruškuc, and Vatter [[@albert:geometric-grid-: Theorem 3.2]]{} showed that ${\operatorname{Grid}}(M)={\operatorname{Geom}}(M)$ if and only if the row-column graph of $M$ is a forest (in this case we say that $M$ *is* a forest). As it happens, all gridding matrices encountered in this paper are forests.

Geometric grid classes are especially tractable because their elements can be encoded by words over a finite alphabet, and for the rest of this section we describe this encoding and its properties. We say that a ${0/\mathord{\pm} 1}$ matrix $M$ of size $t\times u$ is a *partial multiplication matrix* if there exist *column and row signs* $$f_1,\ldots,f_t,g_1,\ldots,g_u\in \{1,-1\}$$ such that every entry $M_{k,\ell}$ is equal to either $f_kg_\ell$ or $0$. It is not hard to prove that every geometric grid class is equal to ${\operatorname{Geom}}(M)$ for a partial multiplication matrix $M$, and this is especially trivial for forests.

[ccccc]{}

(0,0)(100,100) (35,0)(35,100) (0,45)(100,45) (0,0)(98,98) (10,20)[0.04in]{} (20,80)[0.04in]{} (30,60)[0.04in]{} (40,40)[0.04in]{} (50,30)[0.04in]{} (60,50)[0.04in]{} (70,10)[0.04in]{} (80,70)[0.04in]{} (90,90)[0.04in]{}

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(0,0)(100,100) (0,0)(100,100) 