Gov. Robert F. McDonnell, swooping into Fairfax County on Friday to cut the ribbon at Northrop Grumman’s new corporate headquarters, was also scheduled to headline a $500-a-head fundraiser for Supervisor John C. Cook (R-Braddock), Cook’s campaign said.

The governor’s quick political visit, wedged between other duties, underscored the importance of the bellwether Braddock District race for Republicans, who now hold three seats on the 10-member Fairfax County Board of Supervisors.

“Of course, what the Democrats want to do is turn Fairfax County into Arlington,” Cook said Friday.

Ten of Braddock’s precincts are in territory belonging to state Sen. David W. Marsden (D-Fairfax) and thus figure into the GOP’s effort to wrest control of the Virginia Senate.

But the fundraising was to come after business of another sort. Virginia scored an enviable coup by luring Northrop Grumman with at least $12 million in incentives after the Fortune 500 company decided to relocate to the metropolitan region from California. The move bolstered McDonnell’s claim as a business-friendly governor and won him some bragging rights against Gov. Martin O’Malley, his Democratic counterpart in Maryland, and the Disrict, both of which also bid for the company.

But Board of Supervisors chairman Sharon S. Bulova (D) has also claimed some credit. At a recent debate hosted by the Fairfax County Chamber of Commerce, Bulova said the county’s highly regarded schools, parks and other amenities were enough to win over the company without any county money on the table.

Her Republican opponent in the chairmanship contest, Supervisor Michael J. “Spike” Williams, suggested that she was taking too much credit.

After Friday’s ribbon-cutting--which Sen. Mark Warner (D) also attended -- McDonnell was expected to help raise as much as $5,000 per patron at a fundraiser for Cook, who is battling for reelection against Democratic challenger Janet Oleszek.

The most recent campaign funding reports gathered by the Virginia Public Access Project showed that Oleszek, a former School Board member, had outraised Cook in the August reporting period, though Cook had more money overall.

Oleszek wondered whether Cook agreed with the governor’s cost-cutting in public education.

“Governor Bob McDonnell has spent his first two years in office cutting state support for public education and making local taxpayers in Fairfax pick up the bill,” Oleszek said in a written statement.

Oleszek, who has received the endorsement of the Fairfax Education Association, also said the governor appeared to be hinting of further cuts in his remarks Thursday to the State Board of Education.

The Roanoke Times reported that McDonnell told the education board that although education was a “top priority,” the state still faces significant “budget challenges.”