I'm thinking more locally here so pretend that each corner comes from a separate board position (yes, I realize that the overall board position matters a lot). I've been educated that these enclosures are all big moves but how would white go about approaching/reducing so that the black positions remain as small as possible?

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[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . X , X . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

_________________
Thinking like a go player during a game of chess is like bringing a knife to a gun-fight. Thinking like a chess player during a game of go feels like getting knifed while you're holding a gun...

The two on the left would probably not be reduced on a local level. They would only be approached as part of a global strategy. Even locally the choice in all four corners depends on the global position. I have marked some of the possible points with a. That marked b are also possible in special situations. You will need to look up the sequences in SL. This list is not exhaustive, but understand these before looking elsewhere.

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[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . b . . . . |
$$ | . . a X . b . . . . . . . . . a a . . |
$$ | . . b , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . a . . . . . . . . . . . . . . . |
$$ | . . X . a . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . X , X b . |
$$ | . . . X . . . . . . . . a . . a . . . |
$$ | . . . a . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

_________________
Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

If we want to know about the situation without local stones, let's exclude them. Using kombilo on the 90,000 games in my database, the top ten choices by White faced with an otherwise open quadrant and a...

Small-knight enclosure from 3-4 point

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[go]$$B Continuations
$$+ - - - - - - - - - - - -
$$| . . . . . . . . . . . .
$$| . . . e . . . . . . . .
$$| . . . X . . . . b a d h
$$| . . . , . . . . i c . .
$$| . . X . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . j . . . . . . . . .
$$| . . g , . . . . . , . .
$$| . . f . . . . . . . . .
$$| . . . . . . . . . . . .
$$[/go]

Statistics:
16311 matches
Wa: 6262 (Note the meaning of this is that out of 16,311 occurrences of this position, the next play in the region of the board shown was a White play at 'a' 6,262 times)
Wb: 3730
Wc: 1133
Wd: 697
We: 550
Wf: 442
Wg: 400
Wh: 342
Wi: 293
Wj: 264

High enclosure from 3-4 point

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[go]$$B Continuations
$$+ - - - - - - - - - - - -
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . X . . . . c a e i
$$| . . . , . . . . . d . .
$$| . . . X . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . j . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . f . . . . . . . . .
$$| . . b g . . . . . , . .
$$| . . h . . . . . . . . .
$$| . . . . . . . . . . . .
$$[/go]

Statistics:
5002 matches
Wa: 1553
Wb: 787
Wc: 466
Wd: 448
We: 440
Wf: 237
Wg: 222
Wh: 108
Wi: 93
Wj: 93

Large-knight enclosure from 3-4 point

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[go]$$B Continuations
$$+ - - - - - - - - - - - -
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . X . i . . b a d f
$$| . . . , . . . . . c . h
$$| . . . . . . . . . . . .
$$| . . X . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . j , . . . . . , . .
$$| . . e . . . . . . . . .
$$| . . g . . . . . . . . .
$$[/go]

Statistics:
3473 matches
Wa: 1518
Wb: 585
Wc: 360
Wd: 256
We: 130
Wf: 100
Wg: 57
Wh: 50
Wi: 39
Wj: 38

Small-knight enclosure from 4-4 point

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[go]$$B Continuations
$$+ - - - - - - - - - - - -
$$| . . . . . . . . . . . .
$$| . . . f g . . . . . . .
$$| . . . . . . . . b a i d
$$| . . . X . . . . . e . .
$$| . . . . . . . . . . . .
$$| . . X . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . . . . . . . . . .
$$| . . . c . . . . . , . .
$$| . . h j . . . . . . . .
$$| . . . . . . . . . . . .
$$[/go]

Statistics:
3003 matches
Wa: 1169
Wb: 454
Wc: 125
Wd: 119
We: 116
Wf: 103
Wg: 100
Wh: 91
Wi: 82
Wj: 79

Do we see a pattern here?
In addition, these are the results when White plays next. In all cases White plays next more often than Black. However, in all four cases Black sooner or later plays next more often than White plays 'a' (the most frequent White response). Indeed, in all cases the most frequent immediate play is tenuki.

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21