Various methods of ellipsis of part or all of a clause. To these may be added node raising (left and right) and null complement anaphora.

Two clauses can be combined with either coordination or subordination:

Mary mends the gearbox while John practises ballet steps.
Mary mends the gearbox more skilfully than John practises ballet steps.
When a noun phrase appears in both halves, it can be replaced by a pronoun:
Mary mends the gearbox while John practises paradiddles on it.
Mary mends the gearbox more skilfully than she plays the drums.
When the same verb phrase appears in both halves, it can be replaced by (in English) the inflected form of the verb 'do', or 'do so'. The verb phrase is the main verb plus all its baggage:
Mary mends the gearbox and John does so too.
Mary mends the gearbox more skilfully than John does.
Gapping, stripping, sluicing, and the rest are the strategies for substitution depending on how much of the verb phrase is repeated. Verb medial gapping just omits the common verb and keeps material on both sides of it:
Mary bought a new gearbox, John a tutu, and Gritchka a stuffed mole.
(= Mary bought a new gearbox, John bought a tutu, and Gritchka bought a stuffed mole.)
Coordination can be between noun phrases or between verb phrases:
Mary bought a gearbox, a hammer, and a pair of pliers.
Mary bought a gearbox, sanded it down, and installed it in the car.
Left node raising is when a group of words occurs in common on the left, but without being a natural constituent such as noun phrase or verb phrase. It can nevertheless be held in common for the ellipsis:
Mary bought herself a gearbox and John a tutu.
(= Mary bought herself a gearbox and Mary bought John a tutu.)
Right node raising is when a constituent is in common on the right of two clauses. The first copy is omitted and the second serves as the constituent in both clauses. This might involve marking it off by intonation so that it doesn't seem to belong just to the second:
Mary bought and installed a new gearbox.
= Mary bought, and installed, a new gearbox.
(= Mary bought a new gearbox and Mary installed a new gearbox.)
Stripping is when only a single constituent of the second clause is different, so it is left in, and everything that's in common is stripped out. Typically this is used in a context where the one different thing is being contrasted or given focus, so the stripped clause will also have a focal particle like 'not', 'but', or 'too':
Mary is working on the gearbox, but not John.
Mary is concentrating on the gearbox, not the wheels.
Mary has left oil in the kitchen, and in the dining room too.
Sluicing is when the entire verb phrase is omitted, leaving only a subordinate question word in initial position.
Mary saw someone had dirtied the carpet, but didn't know who.
Mary wants to paint the gearbox green, but can't say when.
Null complement anaphora is when the complement of a verb, typically of some kind of saying or thinking, is completely omitted.
Mary wanted help oiling the gearbox, but John refused.
Mary painted the gearbox purple, even though John didn't approve.
Mary told John to move out of the way, and John told the cat.
When we hear a sentence with an ellipsis we know how the missing parts must be supplied for the sentence to make sense and be grammatical. This forced choice in parsing is called reconstruction.

Taxonomy of VP-anaphora taken from Y. Huang, 2000, Anaphora: A Cross-linguistic Study, OUP