Rudolph and Bairstow secure draw
Yorkshire 322 for 9 dec and 225 for 4 v Hampshire 498 for 6 dec
Yorkshire managed to hold out for a draw at Scarborough after a Hampshire declaration on 498 for 6 had put them 176 runs in front with a minimum of 71 overs remaining on the fourth day of the County Championship match.
Although the visitors tried hard to achieve an unlikely victory, Yorkshire were never put under any undue pressure and the game came to a close when they were able to declare on 225 for 4.
The result was still a big disappointment for Yorkshire, who were only able to pick up seven points. Although they remain in third place in the table they may now need to win both of their remaining matches - against leaders Nottinghamshire and Kent - if they are to snatch the title.
In contrast, Hampshire will feel quite happy travelling home with 11 points which were sufficient to move them further away from any relegation fears. Jimmy Adams and James Vince had already put on 215 for the fourth wicket when they resumed their partnership on the final morning with Hampshire on 367 for 3 and leading by 45.
They continued to exact a heavy toll on the bowling, both batsmen striking sixes off Adil Rashid, and Vince swept the leg-spinner for four to raise the 400 in the 106th over and bring Hampshire a fifth batting bonus point.
The stand overtook the previous record for the fourth wicket at North Marine Road of 258 between Phil Robinson and David Byas for Yorkshire against Kent in 1989 and it went on exceed the previous Hampshire highest for the fourth wicket of 263 by Roy Marshall and David Livingstone against Middlesex at Lord's in 1970.
It had reached a massive 278 when opener Adams finally departed for 196 from 343 balls with 27 fours and a six after occupying the crease for 113 overs. He failed to connect while trying to hit Steve Patterson through midwicket and lost his middle stump.
There was more pain ahead for Yorkshire, however, as Vince and Sean Ervine stepped on the accelerator pedal by rattling up 63 runs in eight overs, Ervine driving Patterson over long-on for six and Vince clearing the West stand when Rashid bowled him a long hop.
Vince was in with a shout of converting his maiden century into a double-century until he attempted to hit a second consecutive six off Rashid only to hole out to Jonny Bairstow on the midwicket boundary for 180 - which came from 205 balls with 24 fours and three sixes.
Hampshire were 493 for 5 and five runs later Ervine was run out and they promptly declared, leaving Yorkshire six overs to face before lunch which Lyth and Rudolph safely negotiated.
Runs came swiftly after the interval but Dominic Cork, who had dismissed Lyth in the first innings, continued to pose him problems and at 87 he fended a ball to short leg where Adams scooped up an excellent catch. Rudolph went to his 50 off 74 balls with nine boundaries but then cut David Balcombe straight into the hands of Michael Carberry at point and Yorkshire were 129 for 2 and still trailing by 47.
Anthony McGrath's departure to a slip catch by Neil McKenzie off left-arm spinner, Danny Briggs, raised the visitors' hopes slightly but skipper Andrew Gale and Bairstow saw it through safely to tea.
Briggs had Gale snapped up at leg-slip by Chris Benham in the evening session and the game could still have gone Hampshire's way if Gerard Brophy had not been dropped at first slip by Vince before he had scored.
That would have left Yorkshire effectively 15 for 5 but there were no further alarms and Bairstow just had time to reach his 50 from 71 balls with eight fours and a six before the sides decided to call it a day.