For example: "indisputably-accurate"; "the quickly-ran event"; "the truck-driving man"; "the under-slept woman"; "the power-possessing orb".
What is the term for the words that surround the hyphen?
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In the first two cases:
the only word is "incorrect". Never use a hyphen between an adverb of manner ending in /-ly/ and the word that follows. You can find that rule in any manual that explains how to use hyphens.
In the next two cases:
the word is "hyphenated".
In the final case:
the word is "What does this mean?" Should this be "sleep-deprived"?
The examples in which the hyphen is used appropriately are called compound adjectives (or compound modifiers). The hyphenated phrase serves to describe a noun. As everyone else has pointed out, don't use hyphens with -ly adverbs.
This term does not describe all hyphenated phrases, however. Test-drive is a transitive verb; pattern-seeking is a gerund.
I found this at http://grammar.about.com/od/c/g/compadjterm.htm and thought it might be germane to the topic.
"Interestingly, hyphenation is also used creatively to indicate that an idea that would normally be expressed by a phrase is being treated as a single word for communicative purposes because it has crystallised in the writer's mind into a firm, single concept. Thus, for example, the expression simple to serve is normally a phrase, just like easy to control. But it can also be used as a hyphenated word as in simple-to-serve recipe dishes (M&S Magazine 1992: 9). . . .
"But for creative hyphenation you are unlikely to find anything more striking than this: [2.3] On Pitcairn there is little evidence of the what-we-have-we-hold, no-surrender, the Queen's-picture-in-every-room sort of attitude. (Simon Winchester in The Guardian magazine, 12 June 1993: 27; italic added to highlight the compounds)" (Francis Katamba, English Words: Structure, History, Usage, 2nd ed. Routledge, 2005)
"Adverbs that do not end in -ly may take the hyphen to form a compound adjective. The reason is obvious. A fast-moving script suggests a roller coaster plot while a fast moving script might have pace but it is emotionally charged (i.e., emotionally moving) at the same time." (Bruce Grundy, So You Want to be a Journalist? Cambridge University Press, 2007) Also Known As: phrasal adjective