The algorithm is an algorithm in the sense of $Lax185574.Algorithms$, one for all sentences. For each sentence there are the radius $r$, the depth $d$, the degree $b$ and the constant $c$ of the time bound. The input is the triple of G, a script of the construction sequence and φ, with the encodings of $Lax185574.Encodings$. The construction sequence is the run of the script, see $Lax185574.ConstructionSequences$, so λ is the length of the run. K is the width of $Lax185574.Width$.

The time bound λ · n² · poly_φ(K) is spelled out as $λ · (n+1)² · (K+1)^b$. The graph without vertices has construction sequences of every length, and the factor (n+1)² keeps the bound large enough to read them. The paper says that r and d are computable from φ. Here they only exist. The algorithm is correct on every construction sequence, whatever its width. So an algorithm that needs r and d can try all pairs in parallel, and the right pair costs only a constant factor that depends on φ.