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The Hampstead star, 25, toppled former world No 19 Vekic 6-4 6-3 in the first round to kickstart her Midlands campaign in style.
Dart soared up to a ranking of world No 99 in March and while she has slipped out of the top 100 in recent weeks, the Londoner knows a run deep in Nottingham can catapult her back up the standings.
Asked if she was targeting a return to the world’s top 100, the British No 3 said: “Yes, for sure. At the end of the day, I want to win matches week in, week out and then the rankings will take care of themselves.
“For me, it’s just more of a focus on each match as it comes.“I don’t really have too many goals for this grass court season – I want to enjoy each match, put in good performances, win against each player and try to do it each match.
“Clay is not my best surface, and that’s something I hope to improve on.
“But I know I can play a really good level on grass – it doesn’t take much to swing a match and today has definitely given me some confidence.
“I knew I had to bring my A-game today and in patches I did play really well.
“I’ve definitely made strides in the right direction – I’ve had a little bit of a rough period recently but it’s great to get a win on the board on home soil.”
Dart came flying out of the blocks against Vekic – the current world No 90 – to take the first set 6-4.
And she carried on where she left off in the second to complete a comfortable straight sets triumph with a 6-3 victory.
Dart toppled reigning US Open champion Emma Raducanu, 19, in the last 64 of the Nottingham event last year before the Bromley ace embarked on a remarkable rise to the Grand Slam summit.
And speaking about her compatriot’s journey, Dart - who is on the LTA’s Pro Scholarship Programme, the highest level of support for players aged 16-24 – added: “She’s had an amazing run.
“All of us British girls practice together so I was aware of her – I was always expecting a tough match last year but I managed to come through.“What she’s been able to do over the last year has been remarkable.
“I think anything is possible - anyone can play a good level at any given day and beat anyone. Consistency is the key for rankings.
“Everyone can play good tennis – it’s about doing it every match. Emma has had an amazing year.”
For the latest action on the British summer grass court season, check out the LTA website.