Graham took three of the first four frames and then compiled the highest break of the match, 77, in the fifth on his way to a 5-1 victory.
Two other former world champions, Steve Davis and Joe Johnson, enjoyed better fortune. Davis overwhelmed his English rival Les Dodd 5-0 and Johnson, who succeeded Taylor as world champion, routed Tony O'Connor 5-2.
The rookie Steve Lee continued his fine run with a 5-2 victory over Steve James, the world No 10. The 17-year-old Lee, who reached the quarter-finals of last week's European Open, won the last four frames to set up a match against the New Zealander Dene O'Kane for a place in the televised stages.
WICKES HOME IMPROVEMENT BRITISH OPEN (Derby) Third round: P Ebdon (Eng) bt G Wilkinson (Eng) 5-4; J Johnson (Eng) bt A O'Connor (Irl) 5-2; D Morgan (Wal) bt P Davies (Wal) 5- 1; A Hamilton (Eng) bt D Mountjoy (Wal)5-1; S Lee (Eng) bt S James (Eng) 5-2; S Davis (Eng) bt L Dodd (Eng) 5-0; D Finbow (Eng) bt P Francisco (SA) 5-1; M Hallett (Eng) bt C Small (Sco) 5-3; I Graham (Eng) bt D Taylor (N Irl) 5-1; J Swail (N Irl) bt A Robidoux (Can) 5-2; J Parrott (Eng) bt A Higgins (N Irl) 5-0; J Wattana (Thai) bt M Davis (Eng) 5-4; T Drago (Malta) bt J Wych (Can) 5-2.