Labelling a vertex by the least vertex reachable from it makes the output a function of the graph, so the statement is about computing a function and needs no convention for choosing representatives. Any other canonical choice would do; what matters is that the answer is determined, since a program that may return any of several correct answers would be a weaker claim dressed up as this one.

The least vertex is the infimum of the set of numbers of vertices reachable from $v$. That set contains $v$ itself, so the value is a genuine minimum and the convention $sInf ∅ = 0$ for natural numbers is never exercised. The labelling of the whole graph is the list of these values in vertex order, so its length is the number of vertices.

The order of quantifiers is the content of the theorem: the program and the constant come first, the graph next and the word length last, so one program with one constant serves every graph at every word length. Quantifying the program before the word length is what makes it an algorithm rather than a family of them — a program chosen after $w$ could hide an arbitrary amount of information in its literals — and it is the strong form of uniformity the model supports, since a program can measure $w$ for itself.

The bound is linear in the length of the input word — the number of entries actually handed to the machine, namely $3 + n + 2m$ — which is the input size in the sense the model charges for. Reading the input alone takes that many steps, so the bound is tight up to the constant. The $+ 1$ only keeps the bound from being vacuous on inputs of length 0, of which there are none valid.

One constant does both jobs. An encoding is admissible at word length $w$ when $c * (|x| + 1) ≤ 2 ^ w$, that is, when $2 ^ w$ is at least the very number of steps the claim allows; this is the "the word is wide enough for the input" hypothesis of the word-RAM literature, written out as an explicit inequality against $2 ^ w$ rather than through a logarithm. Nothing further need be asked of the entries: an encoding's entries are vertex numbers, offsets and the two header numbers, all smaller than its own length, so a word length that admits an encoding also holds every number in it.

The fitting condition is a condition on the admissible inputs and not a hypothesis of the claim, because as a hypothesis it would be empty. A graph with an edge has encodings of every length, since a block may list a neighbour repeatedly, so no word length accommodates all encodings of a fixed graph at once and "if every encoding of $G$ fits into a word" would never be satisfied. Restricting the inputs instead says what is meant: at every word length, every encoding that fits is computed within the bound.

Only encodings of $G$ are admitted as inputs; the program may behave arbitrarily on words that encode nothing, and on words too long for its word length.