The word RAM of the archive reads and writes lists of natural numbers, so this module fixes how a graph and a nice tree decomposition are written as one.

A graph on the vertices $0 … n - 1$ is written as $n$ followed by its adjacency matrix row by row, each entry $1$ for adjacent and $0$ for not: $1 + n * n$ numbers. The matrix rather than an edge list, so that a program which fills a table indexed by pairs of vertices reads an entry in one instruction; the length of the word is then quadratic in the number of vertices, and the running times below are polynomial in it, not linear.

A nice tree decomposition is a word $N$ followed by $N$ records of three numbers, one for each node, in an order in which every node comes after its children: the kind of the node ($0$ leaf, $1$ introduce, $2$ forget, $3$ join), a vertex (for an introduce or a forget node) and a node (the second child of a join node). The first child of a node that is not a leaf is the node just before it, and the last node is the root. Bags are not written: they are determined from the leaves upward, a leaf having the empty bag. Listed in this order, the nodes describe a rooted tree in which each node has at most two children. $NiceDecomposition G w D$ says that $D$ describes a tree, that the bags it determines cover the vertices and the edges of $G$, that they are connected as required, and that they have at most $w + 1$ vertices. It is the archive's tree decomposition written out for words.

The root is not required to have an empty bag: every consumer of a nice decomposition only uses that each vertex is introduced once along each path and forgotten after its last edge, and the root's bag can be emptied by forget nodes at the cost of a linear number of further nodes. A word with a nonempty root bag is still a nice tree decomposition here.