First I should want to draw attention to the similar endgame
Timofeev-Hi with an additional pair of rooks.
http://www.chesspub.com/cgi-bin/yabb2/YaBB.pl?num=1201632403 John Cox himself commented here, that it was a mistake by Parker to swap rooks.
I appologize to John, for not posting comments on his analysis yet. I will do so probably on sunday night. My usual approach to this kind of endgame is to study some similar but simplified position, that function as a sort of sign-posts. In this particular endgame I started to look for for various fortresses with the same pawn structure and material balance.
As they are indeed quite simple, they probably will be not very usefull for John, but might enlighten some other readers.
In general white can expect to win this endgame if (a) he can play his bishop to queenside behind black’s pawns and win some of the pawns.
(b) if he can create a passed pawn on the kingside, supported by his king on 6th rank
(c) if he can swap bishops, as the king+pawns endgame is a relative easy win.
Black’s task is to deny the white pieces access to his position. In other word he has to erect a fortress. My own investigation of this endgame has not started with examining trees of variations, but by researching some (simplified) positions and looking for fortresses.
In this posting I will present some fortresses, and some positions that are “not quite a fortress” (perhaps Flear would call them NOQAFs).
White has to take some care, not to help black building his fortress. Especially if he puts pawns on squares the colour of his bishop. Here is a first example:
diagram 1 
White has incautiously put his pawn on g5. Effectively denying access to the all important d8 squares d8 and c7.
If black leaves his king at g6, the white king cant get in as well, nor can it help to advance the g-pawn.
Black will just bide his time playing Bd8-c7-d8
diagram 2 Here g5- is barricaded by a black pawn. Since white cant approach this pawn with his king, this obstacle serves as well to keep white’s bishop out. Note that the white king cant help his passed e-pawn.
Black just moves his bishop between e7 and d8, since this bishop now has the task to defend g5.
diagram 3 Here we see again a fortress build with the aid of an extra white pawn on the black squares.
White cant make any progress if black keeps playing Bd8-c7-d8.
when is a fortress not a fortress? An essential characteristic in the first three positions is that black has sufficient waiting moves, that don’t alter or worsen his position. For example if we put another pawn on c7 in the last position, then suddenly black is out of waiting moves and must play either Kf8 allowing Kg6-f7-Bh6+ or play 1…b5 2.b3, b4 3.Be3 and black again is out of moves.
The situation where black has the move and must worsen his position is called Zugzwang.
diagram 4 Here is another zugzwang example.
Note that black’s bishop is paralysed, it has nowhere to go. If the black bishop had been on d8 instead of e8 this example would have been drawn.
1.Be1!, and black is out of moves:
a) 1…b5 2.Bxa5
b) 1…Ke8, 2.Kg6!,Kd7 3.Kf7
c) 1….Kf8, 2.Kg6!,Kh8 3.Bh4,Kg8 4.f7+,Kh8 5.Bf6+
So zugzwang is one method of breaking a semi-fortress. Another method is sacrificing material to force an entry into black’s castle. diagram 5 Here waiting gets white nowhere (1.Kh4,Kh6 2.Kg4,Kg6).
White has to sac his f-pawn to make progress.
.1.Kf4,Kh6 2.f7, Kg7 3.Kg5,Kxf7 4.Kh6,b5 (otherwise whites king gets to g6 and wins the e6 pawn)
5.Kh7!, b4 6.Kh6 and white wins. Note 1, 3.f8Q+ is overdoing it, as after
3…Kxf8 4.Kg5,Kg7 black has a new fortress.
Note 2: 5.axb5?,cxb5 6.cxb5,Ke8! and black wins as he can force a breakthrough with
7…c4 8.bxc4 a4 and his a-pawn cant be stopped.
diagram 6 
John Cox allready pointed out in his analysis that this type of position is a fortress.
White cant restrict black’s king moving between g4 and g3 and at the same time prevent black moving his bishop. On top of that white has to prevent black gaining the f5-square for his king.
Just place white’s bishop on b6, where it restricts its black college, and it has no control of g3.
Place it on f4 and black bishop is again freed.
diagram 7 In this position again white’s pawns on the black squares help the defender. A pawn break with f5 isn’t possible as after
1.f5+?,exf 2.Kd3,Kf7 it is white who is lost.
finally a fun position that drives home the lesson on the importance
of access points. diagram 8 In the last diagram black keeps the draw despite his material deficit.
White’s only hope for victory is a pawn break on the queenside, as the kingside is effectively barricaded.
White has to sacrifice if he wants to play for a win.
1.b4,cxb4 2.c5 (if white delays this move, black will play 2…c5 himself closing the gates.)
2….b5! (other moves give white a good chance to break through)
3.Kb3,Kc7
4. axb5, cxb5 diagram 9 White has to play c6 sooner or later to make progress
line 1 5. Be1, Kb7 (5..Kc6?? 6.Bxb4 and white wins) and black waits playing Kc7-b7-c7
line 2 5. c6!?, Kxc6 6.Ba7, (at last the bishop has entered the fortress)
6... , Kd5 and now white has another problem: the defence of his weak d-pawn.
Again the position is drawn.
Some sample-lines after
5.c6,Kxc6 6.Ba7,Kd5: line 2a 7.Bb6,a4!+ 8.Kxb4,Kxe5 9.Be3, Kf5 black draws by deflecting the bishop with his a-pawn:
10.Kxb5,a3! 11.Bc1,a2 12.Bb2,Kxg5 drawn.
line 2b 7.Bb8, Kc5 8.Bd6+,Kd5 white cant make progress either. If he moves his king to defend the e-pawn.
9.Kc2,Kc6 10.Kd3 then black starts moving his queenside pawns:
10…b3! 11.Kc3,a4 and now it turns out that 12.Kb4 is not on the menu because of 12…b2!
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