Detective Chief Inspector Hamish Campbell, the man leading the inquiry, said a bus driver has come forward since an E-fit picture of the suspect was released.

The bus driver believes that he took the man four stops from Fulham Palace Road to Putney Bridge tube station, where police inquiries are now being focused.

Police are keen to hear from anyone who may have seen Ms Dando on the morning she died, because a gap exists in their knowledge of her movements from about 1100 to 1130 BST.

Witnesses have come forward since police released the E-fit and pictures from security cameras showing Ms Dando on a shopping trip shortly before she died.

The Range Rover seen speeding down Fulham Palace Road

On Tuesday they took to the streets around her home to quiz more people, and pulling all those different strands together a picture has been built up of the sequence of events before, during and after Ms Dando was shot in the head on Monday, 26 April.

10.05 - Ms Dando leaves her fiance's house by car.

10.23 - She stops for petrol at a BP garage on the Great West Road.

10.46 - She is seen at Rymans stationers at the Hammersmith shopping mall.

10.55 - Ms Dando is spotted at Dixons Link electrical shop. She also bought two fillet Dover sole fish and police are trying to track down the shop where she got them from. She received and made a number of calls during her shopping trip.

10.40 - A window cleaner working at No 26 Gowan Avenue sees a man with light coloured hair, smartly dressed in a light grey two-piece suit talking into a mobile phone. He was loitering outside Ms Dando's house at No 29.

About 11.30 - Ms Dando is murdered on her doorstep by a single bullet from a 9mm semi-automatic pistol. Screams were heard but no gunshot. Two witnesses saw a man running from No 29. He was said to have thick black hair and be wearing a waxed jacket. At a similar time a man was seen standing at the junction of Gowan Avenue and Munster Road. He was wearing black-framed glasses and was in an agitated state looking down the two roads. He was said to be in a dark-blue pin-striped suit.

Shortly after 11.30 - A motorist is forced to brake sharply while driving along Fulham Palace Road when a smartly dressed man runs out in front of him. He described the man as about 5'11", with short dark hair and dressed in a dark suit. Also, two witnesses saw a man near Bishops Park using his mobile telephone and hanging an arm over the park railings.

11.45 - DCI Campbell says a man resembling an E-fit of the suspect was seen getting on a bus. He was reportedly sweating heavily and speaking into a mobile phone. The bus took him from Fulham Palace Road to Putney Bridge tube station where police are now concentrating their inquiries.

Another witness described a man "running for all his life" from near Ms Dando's home on Gowan Avenue to Bishop's Park. The man reportedly was almost hit by a van on his way across Fulham Palace Road.

11.52 - A picture from a CCTV camera shows a metallic blue Range Rover on Fulham Palace Road travelling south at high speed. The car was seen leaving Doneraile St opposite Gowan Avenue. It had one person in it and it leaned over as it swept into the main road.

The driver was seen running a red light before driving south and was filmed by the camera at Putney Bridge.

Police said the man at the bus stop could not have been the same person as the one in the Range Rover which may indicate Ms Dando's killer had an accomplice.