Facing next year’s congressional elections minus three incumbents, Republicans are doing themselves no favors by staging a bitter primary for a fourth U.S. House seat next Tuesday.

Steve Buehrer and Bob Latta are waging a mud-soaked campaign for the seat that belonged to Paul Gillmor, who died Sept. 5 in an apparent fall down the stairs in his suburban Washington home. The winner will run Dec. 11 for the seat that’s been Republican for decades.

The outcome of that election will provide a compass for 2008’s congressional contests. Reps. David Hobson of Springfield, Deborah Pryce of suburban Columbus and Ralph Regula of the Canton area already have announced they won’t seek re-election.

Hobson’s district is considered safe for the Republican candidate, but Regula’s and especially Pryce’s will take strong — and expensive — campaigns to stay in the GOP column. Democrats are expected to also charge hard for the seats held by Republ-icans Steve Chabot of Cincinnati, who got 52 percent of the vote in 2006, and Jean Schmidt of Loveland, who won by fewer than 3,000 votes.

Republicans have controlled the Ohio delegation since the GOP sweep of 1994. The GOP now has an 11-7 majority in Ohio, so a swing of three seats would give Democrats control. Republicans have said they will make a serious run at Democrat Zack Space of Dover, who easily took the eastern Ohio seat held by Bob Ney, now imprisoned after a bribery scandal.

Recent polling by Quinnipiac College in Ohio found Democrats leading Republicans in each possible presidential matchup, and Republicans are still smarting from last year’s near sweep by Democrats of statewide offices the GOP had held for more than a decade.

The Democrats’ best chance is in Pryce’s central Ohio district, which has been in the GOP column for decades. However, Pryce won last year only after a recount and the loser, Mary Jo Kilroy, is a lock for the Democratic nomination next year. Republicans are still looking for a candidate after their two strongest names, former Attorney General Jim Petro and state Sen. Steve Stivers, turned party officials down.

Kirk Schuring, a state senator from Canton, is the likely GOP candidate for Regula’s northeast Ohio seat. Democratic state Sen. John Boccieri of New Middletown has the field to himself for the Democratic nomination.

In southwest Ohio, Democrat Victoria Wulsin of Cincinnati is repeating her run against Schmidt and Democrats are confident that state Rep. Steve Driehaus, also of Cincinnati, can finally topple Chabot. Republicans are clearing the field for Hobson’s western Ohio seat for state Sen. Steve Austria of suburban Dayton.

Gillmor’s northwest Ohio district is one of the most Republican-leaning in the state, but GOP leaders worry the bruising primary between Buehrer and Latta, both state lawmakers, could give Democrats an opening.

Each has highlighted the other’s vote for the ‘‘Bob Taft gas tax’’ increase, invoking the unpopular former governor’s name. And both have brought Tom Noe into the race, pointing out in ads that the other took campaign money from the imprisoned rare-coin dealer and GOP fundraiser who played a big part in a state Bureau of Workers’ Compensation investment scandal that crippled Republican candidates.

All the vitriol finally led state Rep. Kevin DeWine, the Ohio GOP’s deputy chairman, to send rebukes to both campaigns.

‘‘The direction this campaign is headed will tarnish your reputations and those of your campaign consultants and supporters. But more important than the reputation of any individual, the tenor of the race is poised to harm our party,’’ DeWine wrote.

Both Latta and Buehrer feel the party can survive their primary and defeat the likely Democratic nominee, Robin Weirauch, who is making her third bid for the seat.

Buehrer said the key is making sure people go to the polls when ordinarily they would be holiday shopping.

‘‘A complicator in this race is the very short time frame,’’ he said. ‘‘There will be some challenges in pushing turnout.’’

John McCarthy is a writer for the Associated Press.