Keres Variation has the distance zero to the basic KIA. Slav, Catalan and Double Fianchetto are of height 1 above the basic KIA (corresponding to pawn moves c6, e6 and g6 resp.)
KIA and hierarchy of 3g|5d-openings


Basic KIA stratum, ground zero. I list openings with their Rybka evaluations (chessok.com):
- 1. g3 d5 +0.07
- 1. g3 d5 2.Nf3 +0.07
- 1. g3 d5 2.Nf3 Nf6 +0.07
- 1. g3 d5 2.Nf3 Nc6 +0.11
- 1. g3 d5 2.Nf3 Bg4 +0.15
- 1. g3 d5 2.Nf3 Bf5 +0.18
- 1. g3 d5 2.Nf3 Bd7 +0.18
- 1. g3 d5 2.Nf3 Nd7 +0.22
- 1. g3 d5 2.Bg2 -0.04
- 1. g3 d5 2.Bg2 Nf6 +0.07
- 1. g3 d5 2.Bg2 Nc6 +0.11
- 1. g3 d5 2.Bg2 Bf5 +0.29
- 1. g3 d5 2.Bg2 Bg4 +0.56
... to be continued

Basic KIA stratum, 5 plies.
- 1.Nf3 d5 2.g3 Nd7 3.Bg2 +0.22
- 1.Nf3 d5 2.g3 Bf5 3.Bg2 +0.18
- 1.Nf3 d5 2.g3 Bg4 3.Ne5 +0.15
- 1.Nf3 d5 2.g3 Bg4 3.Bg2 +0.11
- 1.Nf3 d5 2.g3 Nf6 3.Bg2 +0.07
- 1.Nf3 d5 2.g3 Bf5 3.Nc3 +0.04
- 1.Nf3 d5 2.g3 Nc6 3.Bg2 -0.04
- 1.g3 d5 2.Bg2 Nc6 3.Nc3 -0.11

Basic KIA stratum, 6 plies.
- 1.Nf3 d5 2.g3 Nc6 3.Bg2 Nf6 +0.11 2 knights defense
- 1.Nf3 d5 2.g3 Nc6 3.Bg2 Bg4 +0.11
- 1.Nf3 d5 2.g3 Bf5 3.Nc3 Nf6 +0.11
- 1.Nf3 d5 2.g3 Bf5 3.Nc3 Nf6 +0.11 2 knights attack
- 1.Nf3 d5 2.g3 Bg4 3.Ne5 Bf5 +0.15
- 1.Nf3 d5 2.g3 Nc6 3.Bg2 Bf5 +0.18
- 1.Nf3 d5 2.g3 Nf6 3.Bg2 Bf5 +0.18
- 1.Nf3 d5 2.g3 Nf6 3.Bg2 Nbd7 +0.22
- 1.Nf3 d5 2.g3 Bg4 3.Ne5 Be6 +0.22
- 1.Nf3 d5 2.g3 Nf6 3.Bg2 Bg4 +0.22
- 1.Nf3 d5 2.g3 Bg4 3.Ne5 Bh5 +0.26
- 1.Nf3 d5 2.g3 Bg4 3.Bg2 Nd7 +0.26 Keres variation
- 1.Nf3 d5 2.g3 Nc6 3.Bg2 Nd7 +0.29
- 1.Nf3 d5 2.g3 Nf6 3.Bg2 Bd7 +0.33
- 1.Nf3 d5 2.g3 Bf5 3.Bg2 Nd7 +0.41
King's Indian Attack (A07) 1.Nf3 d5 2.g3
I wanna compare all openings with d5+g3 pawn structure and maybe some additional black pawns on 6th row. I will call it KIA hierarchy or 3g|5d-hierarchy. Here's whats I found in Game Explorer so far.
Hungarian Opening: Slav Formation (A00)
1.g3 d5 2.Bg2 c6
Hungarian Opening: Catalan Formation (A00)
1.g3 d5 2.Bg2 e6
King's Indian Attack: Double Fianchetto (A07)
1.Nf3 d5 2.g3 g6
King's Indian Attack: Keres Variation (A07)
1.Nf3 d5 2.g3 Bg4 3.Bg2 Nd7
King's Indian Attack: Yugoslav Variation (A07)
1.Nf3 d5 2.g3 Nf6 3.Bg2 c6 4.O-O Bg4