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The question is how to write the following sentence in a parenthetical form:

Loops — A loop is an arc that pairs a vertex to itself. — are not allowed here.

Are the following sentences correct, or should they be plural?

Loops, an arc that pairs a vertex to itself, are not allowed here.

Loops (an arc that pairs a vertex to itself) are not allowed here.

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You can't put a sentence complete with its punctuation as a parenthetical phrase, so the capital letter and full-stop are wrong.

Loops — a loop is an arc that pairs a vertex to itself — are not allowed here.

That doesn't read badly; but if you wanted to use commas, which [it seems to me] bind the parenthetical phrase more closely to the rest of the sentence than dashes do, it would need to be plural to match the main subject. Note that the vertex doesn't need to be plural, just the arcs drawn from it.

Loops, arcs that pair a vertex to itself, are not allowed here.

I prefer the first one.

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  • Your first two sentences are incompatible (even allowing for 'parenthetical phrase' meaning 'the expression within parentheses, commas, or dashes' (which is usually also called 'a parenthesis'). The parenthesis 'a loop is an arc that pairs a vertex to itself' is a bona fide sentence (the definitions given at, for instance, thefreedictionary.com/sentence do not mention any punctuation as being necessary). You are right in that a parenthesis which is also a bona fide sentence will drop the initial capital (unless there is a further reason to retain it) and closing full stop.
    – Edwin Ashworth
    Commented Mar 19, 2013 at 14:23
  • @EdwinAshworth That's what I meant: a complete sentence including its punctuation. I've modified things.
    – St John of the Cross
    Commented Mar 19, 2013 at 14:33
  • @Charles itself goes with a vertex, not arcs.
    – St John of the Cross
    Commented Mar 19, 2013 at 15:04
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The difference is in the type of parenthesis employed. In both variants, the parenthesis is a pragmatic marker adding (syntactically) non-essential information, further to the matrix sentence.

The first variant is a complete sentence (whose punctuation, as we have seen in comments, must be modified):

Loops — a loop is an arc that pairs a vertex to itself — are not allowed here. or

Loops (a loop is an arc that pairs a vertex to itself) are not allowed here.

The second variant is an apposition, syntactically equal to the subject of the matrix sentence, and hence needing grammatical agreement:

Loops, arcs that pair a vertex to itself, are not allowed here.

Here, commas as well as parentheses or dashes may be used to offset the apposition; notice that either Loops or arcs that pair a vertex to itself may be dropped without impairing syntax; the remaining noun phrase can fulfil the role of subject quite adequately on its own. One could switch the order of 'loops' and 'arcs that pair a vertex to itself', but then the commas would not be a good option.

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