The radius is at least 1, which is the case used. The chain is produced by a function $chain$ of r, ε and the graph, and the algorithm $A$ computes this function, in the sense of $Lax185574.Algorithms$. The algorithm and the absolute constant $K$ of the length bound come first. On every graph the output is a chain of length at most K · r/ε + K. The algorithm does not depend on the class. For a monadically dependent class 𝒞, a radius r and ε, the constant $c$ bounds the width by c · (n+1)^ε on the members of 𝒞. The input is the triple (r, ε, G) and the output is the list of the sets of the chain. The chain is listed in increasing order, ∅ = S₁ ⊆ ⋯ ⊆ S_m = V(G), as in $Lax185574.Chains$. This is the reverse of the order of the paper.

Deviation from the paper. The paper claims the running time Ô_{𝒞,r}(n³), whose constant does not depend on ε. Here the time bound is O(n^{3+δ}) on members of 𝒞 for every δ > 0, with a constant c depending on 𝒞, r, ε and δ, as in $Lax185574.ChainTheorem$. This is weaker than the paper and is what $Lax185574.DefinableSequences$ needs. A caller chooses δ independently of ε, so that the time fits the word lengths of its own statement. The statement is the companion paper's theorem with the chain reversed. The word lengths at which the statement speaks also depend on the class, through c, as in $Lax185574.DefinableSequences$.