Albion are asking fans not to invade the Amex pitch should the club seal promotion to the Premier League.
The Seagulls are top of the Championship, two points ahead of second-placed Newcastle, and 12 points clear of third-placed Huddersfield.
They could clinch promotion on Easter Monday at home to Wigan and this afternoon, Albion have released a statement asking fans not to encroach on to the pitch after the final whistle.
Club captain Bruno said: "Regardless of the outcome of our final two matches, the players and staff want to say thank you to all of our supporters, young and old, wherever they sit in the stadium, by walking around the pitch after the match. If supporters are on the pitch, this will not be possible."
Manager Chris Hughton added: "I hope all our supporters will respect the wishes of the staff and players - and also our security staff, whose job it is to keep everyone in the stadium safe - by staying off the pitch at all times, including the end of the game, regardless of how our season finishes."
The Albion statement said: "We would like to ask all fans to respect this request of the club, staff and players, and also to work with our safety and stewarding staff at both remaining matches this season.
"Even if we were lucky enough to secure promotion against Wigan on Monday, then our players and coaches would want to aim do everything possible to secure the Championship title.
"With further games to be played after each match, it's vitally important to ensure player safety and avoid unnecessary injuries - for both our own players and our visitors.
"We also want our pitch to be in pristine condition for the visit of Bristol City later this month.
"Furthermore, following the Bristol City match, we still have matches to be played on the Amex pitch, including the Sussex Senior Cup Final, in which our club is taking part. Out of respect to our younger players and Crawley Town, our opponents, we would not want the pitch damaged - particularly as the club's ground staff will not have enough time to repair significant damage to the pitch.
"It's also important for us to point out that it remains an offence to enter the field of play, at any time, and this could result in arrest and exclusion from the stadium.
"If we are in a position to celebrate, and with the prospect of Premier League football to watch, we really do not want to have the occasion spoiled in any way or to be in a position where we may have to issue bans to supporters for next season."
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