With 15 minutes left in an increasingly scrappy match, the Arsenal manager, adhering to principles now fully developed by his long career in charge of the London club, withdrew Paul Merson for Andy Linighan. A defender for a forward and at that moment, Graham announced without saying a word that he had settled for the draw.
Had Graham or Arsenal been of a different nature they might have sneaked a late winner, for by this time they were creating the more clear-cut chances, and Walker might just have been out of a job. A win still eludes him, but a draw is not yet a sackable offence.
Everton began with genuine purpose. Urged on passionately and desperately by a crowd of 32,003 - their second largest of the season - they responded appropriately, although more with passion than desperation.
There was a sense of style about their play, almost as though the crowd - who had given them a rapturous reception - had injected them with confidence. To any neutral observer it was almost touching that so many had refused to desert what had seemed such a patently losing cause.
Everton and their fans deserved the somewhat ungainly goal which arrived in the 14th minute. A free-kick on the right after the bristling Graham Stuart had been tripped made its way along the area. It fell from an eventual deflection to David Unsworth, who turned and managed to hit it with enough power to pass David Seaman.
If only Everton had been allowed to extend their lead two minutes later, the win - already seeming as though it would be a cure-all - might have been theirs. Ian Durrant fled down the right and squared adeptly for Joe Parkinson who shot studiously into the left-hand corner.
The fans' delirium was cut short because the offside flag was raised. It was difficult to see where any interference lay, but Everton are bottom and such decisions afflict clubs in their position. Ten minutes later, Arsenal were level with the sort of goal that also afflicts bottom-placed clubs. Stefan Schwarz was given the ball in space some 30 yards out. He was allowed to dwell on it for far too long, and let fly a hard and swerving shot. Neville Southall might have covered it - he did not.
The self-belief ebbed away and was never to be brought back. It was to Everton's credit at least that they sustained their challenge, but it was not with quite the same conviction. Merson might have twice scored for Arsenal, being sent clear by Kevin Campbell and getting past his marker in the first half, and then being set up from 12 yards by the same colleague in the second.
Merson might be ring-rusty, but such an excuse can hardly be made for Daniel Amokachi, who has scored one goal in 11 games. He had two opportunities to score a second, one from a lovely move between Durrant and Stuart, but while he works hard his anxiety level rises almost visibly when the goal looms ahead.
Graham praised Everton's effort and organisation, but he was clearly not going to be the man to let them have their first victory of the season.
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