Jennifer Boysko, trailing in her race for the Virginia House of Delegates by 56 votes, refused on Wednesday to concede the race against Del. Thomas Rust (R-Fairfax), as Democrats hoped they could pick up what would be a crucial seat.

As several other Democrats conceded their close races Wednesday, party officials said they hoped that the final scrubbing of votes in the 86th District contest could give them 34 seats in the House — the minimum needed to sustain a gubernatorial veto.

Although Rust has declared victory, Boysko said she plans to petition for a state-funded recount if the margin remains within one percentage point after election officials examine results in every precinct to find out whether any counting errors might affect the outcome. Boysko, a former aide to Fairfax County Supervisor John Foust (Dranesville), was behind by 0.27 percentage points in unofficial results posted by the State Board of Elections on Wednesday.

The results also could have profound implications for control of the state Senate, which is divided 20-20 between Democrats and Republicans.

State Sen. Ralph Northam (D-Norfolk) captured the lieutenant governor post, making him the chamber’s tie-breaking vote.

But a special election will be held to fill his seat, as well as one to replace the winner of the attorney general race between state Sens. Mark Herring (D-Loudoun) and Mark Obenshain (R-Harrisonburg) — another too-close-to-call contest that could be headed for a recount.

Democratic and Republican operatives agreed that Northam’s seat, while potentially competitive, leans Democratic, while Obenshain’s tilts strongly GOP.

Herring’s seat, which includes eastern Loudoun and a western slice of Fairfax County, would be the most intriguing of the three. President Obama won 59 percent of the vote in the district in 2012, but conservatives are a strong presence in Loudoun and could put the seat in play — particularly in a low-turnout special election.

Some key Democrats favor Jennifer Wexton, who lost the 2011 race for Loudoun commonwealth’s attorney, to succeed Herring. Herndon Town Council member Sheila Olem (D) also may run.

Across the aisle, 10th District Republican Committee Chairman John Whitbeck is widely considered a likely candidate, with strong connections to conservative activists in Loudoun. He sparked controversy in September when he told an anti-Semitic joke at a rally for Ken Cuccinelli II (R). Whitbeck eventually apologized for the remark.

It’s also possible that Del. Joe T. May (Loudoun) could run. May lost his Republican primary earlier this year to Dave LaRock, a more conservative candidate.

May said in an interview Wednesday that “it’s really too early to make any comment at all” on the possibility of a Senate campaign.

In the House races, staff members for candidates in both parties looked for hopeful signs while watching election workers pore over results tallied on touch screens and tape ballot machines.

The process, known as a canvass, includes counting absentee ballots and provisional ballots handed to voters without proper ID. It isn’t likely to be completed until the weekend, said Brian Schoeneman , secretary of the electoral board in Fairfax County, home to the three House districts most in contention.

“Election night is a very busy night; it’s very hectic and it’s all day long with staff here working up to 15 and 20 hours,” he said. “We want to make sure there are no transcription errors or that nobody accidentally flipped a 2 into a 5” in total tallies.

One such error was detected in a precinct’s tally for the attorney general’s race, leading to 528 votes for Obenshain there instead of 258 votes, Schoeneman said, adding that the mistake was reported to state election officials. But no problems were found in the tallies for House races.

Election officials in Loudoun and Prince William counties reported no errors.

After resisting for most of Wednesday, Democrat Kathleen Murphy, who had been locked in a fierce battle against incumbent Barbara Comstock (R-Fairfax) in the 34th District and was behind by 431 votes, conceded late in the afternoon.

“We may have fallen short, but we’ll keep fighting for sensible legislation that protects all the people in Virginia,” Murphy said in an e-mail statement.

Comstock had declared victory Tuesday night.

In the 13th District, which includes parts of Prince William and Manassas Park, Democrat Atif Qarni also conceded Wednesday after coming within 498 votes of unseating longtime Republican incumbent Robert G. “Bob” Marshall, who has sailed to victory in past elections.

“I’m sure the outcome is not 100 percent accurate, but we’re pretty content with the system,” said Qarni, a former Marine and a middle school teacher in Prince William County.

In the 31st District, which includes portions of Prince William and Fauquier counties, Democrat Jeremy McPike conceded to Republican incumbent L. Scott Lingamfelter, who was ahead by 234 votes. “I called Scott a few moments ago and congratulated him on his win,” McPike wrote on his Facebook page.

Del. David Toscano (Charlotsville), the House Democratic leader, said the state Democratic Party may pursue recounts even in races in which candidates have conceded.

The party is hopeful, he said, about the outcome of the race in the 87th District between Democrat John Bell and freshman Del. David Ramadan (R-Loudoun.)

Bell conceded defeat in that race late Tuesday. But on Wednesday, a campaign spokesman said the retired Air Force officer will reevaluate that decision if it appears that enough uncounted absentee and provisional ballots could make up the difference of 195 votes between him and Ramadan.

Ramadan said he was not worried about that likelihood and expressed glee about being able to survive a race in which Bell challenged his ethics and raised more than $800,000 — nearly twice the amount Ramadan raised.

During his first election, in 2011, Ramadan won by 49 votes and survived a recount that, he said, added just three votes to his opponent’s tally.