By Patricia Zengerle and Susan Heavey

CLEVELAND/WASHINGTON (Reuters) - If there was any doubt that the U.S. presidential fight hinges on Ohio, an awkward campaign airplane traffic jam at the Cleveland airport made it clear on Tuesday.

Minutes after Republican presidential candidate Mitt Romney touched down on the tarmac, Vice President Joe Biden swooped in on Air Force Two for an unannounced - but what the White House said was long-planned - visit.

Biden's move seemed aimed at dampening the challenger's 11th hour visit to the battleground state.

Romney, who was waiting for running mate Paul Ryan to land, stayed onboard his plane until Biden boarded his motorcade and left the tarmac, which became even more crowded when Ryan's plane landed minutes later.

Only President Barack Obama, who is spending the day in Chicago, was missing from the dance of the jets.

The awkward three-way stop underscored the importance of Ohio in a presidential race that polls show is virtually tied.

The Midwestern state is seen as the most critical in the candidates' race for the 270 electoral votes needed to win the presidency. No Republican has ever captured the White House without having Ohio in his column.

"That sums up what is going on today. Everyone is competing for that last vote," Stephanie Cutter, deputy manager for the Obama campaign, told Fox News.

Biden, seeking a second term with Obama, made the stop before flying out to meet the president in Chicago.

From the airport, the vice presidential motorcade headed to the Landmark Restaurant, a Greek diner near Lake Erie, for some last-minute campaigning -- and a quick meal with his wife and family.

Biden mingled with the eatery's patrons before later settling down to a table with his family, according to media accounts.

Twana Matthews, eating at the diner with her fiance Andre McCray, told Biden she had voted for the Democratic ticket as soon as the polls opened at 6:30 a.m. and offered to share her food, a press report of the encounter said.

Biden took a picture of her with his family using her smartphone before moving on as Matthews cheered him on, saying "Biden! Biden!," reports said.

When asked by reporters if he saw his Republican opponents' planes and had any message for them, Biden said: "When we landed they said they were there. That's good," according to press reports.

Biden and his wife, Jill, later headed back to the Cleveland airport, where they posed for pictures on the tarmac still holding the Romney and Ryan planes before boarding for Chicago, they said.

Romney and Ryan later arrived at the Cleveland suburb of Richmond Heights to thank campaign volunteers before heading to a nearby fast-food restaurant for burgers.

At the campaign office, 81-year-old Phyllis Froimson, who retired after a career in finance, said she was counting on a Romney victory.

"I hope so. I've got my hopes on it. I say yes," she said when asked if she thought Romney would win.

At Wendy's, Romney told the women behind the counter he stopped there because the chain began in Ohio.

One restaurant employee said she was going to vote later but would not say who she would pick.

Romney and Ryan then headed back to the Cleveland tarmac to head to their next stops in Pittsburgh and Boston.

Earlier, Romney said he felt "great about Ohio."

Obama also made an unannounced stop in Chicago on Tuesday to thank his campaign workers. His plane - Air Force One, was still on the tarmac when Biden landed Tuesday afternoon.

(Additional reporting by Thomas Ferraro; Writing by Susan Heavey in Washington; Editing by Ros Krasny and Bill Trott)