LAS VEGAS — And they're off! Barack Obama and Mitt Romney galloped out of the post-debate gate Tuesday to woo swing state voters in the final two weeks of a race that seems destined for a photo finish.

Both will criss-cross America as they seek an edge in the presidential contest, but one state in particular -- the Midwestern prize of Ohio -- will get the lion's share of visits.

Senior Romney advisor Kevin Madden said: "People are really getting to that point where they are making their mind up in key states."

Barely nine hours after the third and final debate wrapped up Monday night with both sides declaring victory, Romney was on a plane to Nevada, from where he was to head on to the Rocky Mountain state of Colorado.

Both these states were won by Obama in his 2008 White House victory, but polls suggest they could swing to the Republican this year, paving stones on a road to a possible victory that must almost certainly pass through Ohio.

After his western jaunt, Romney will spend at least Wednesday and Thursday week in Ohio, part of America's industrial heartland, still leaning Democrat in most state polls but somewhere the Republican can sense a new momentum.

"He'll be going to Ohio and Virginia, he'll be back in Florida," said Ed Gillespie, another top Romney advisor. "We're obviously going to those states that are most critical to our path to electoral college success."

If history is a guide, Romney's yellow brick road to the 270 electoral votes needed for victory on November 6 must go through Ohio -- no Republican has ever won the White House without also winning the Buckeye State.

Until this month Obama appeared to have a lock on Ohio and most of the other swing states, posting a double digit lead in some polls and dominating key demographics -- in particular women.

But, after Romney bested a lackluster Obama in their first debate on October 3, the race began to narrow. First in nationwide polls then, crucially, in the 10 battlegrounds that will decide the result on November 6.

He still faces an uphill struggle in Ohio, however.

An average of recent state polls calculated by website gives Obama a 1.9 percent lead, while the polling blog for the New York Times estimates the president's advantage at 2.9 percent.

No polling firm has ever given Romney the advantage in Ohio.

The president is determined not to cede this lead, which could probe his last line of defense against a Romney victory if all else fails. So he was in Ohio again on Tuesday, seeking to build on a better debate Monday.

"Yes, we have been through some tough times, but you have never seen me quit," Obama told crowds in Florida, before heading to Ohio, maintaining the air of confidence he exuded in his final head-to-head clash with Romney.

For weeks the Republicans have tried to convey a similar confidence that their game plan in Ohio will bring victory, hoping an impression of momentum not yet completely borne out by local polling will prove self-sustaining.

Romney political director Rich Beeson recently mapped out the grassroots game plan which has seen volunteers knock on 1.4 million doors and make 3.7 million phone calls in the state since May.

But one of Obama's trusted allies in the Senate, number two Democrat Dick Durbin, insisted the president would hold on to Ohio.

"We're not playing 'not to lose.' We're playing to win," he told AFP, predicting the race would come down to eight states.

Ohio may be leaning for Obama -- although the numbers have tightened -- but Romney is favorite in Florida and Virginia is a dead heat.

If Romney can hold Florida and move the needle in his favor in one of the other two, he has a realistic shot at winning. If he loses both Florida and Ohio, the math just isn't there for a Romney victory.

But with no Republican ever winning the presidency without also winning Ohio, Romney will sink massive human and financial capital into the state.

Studies show the two campaigns have already spent some $100 million on television ads in battleground states in the closing weeks of the race, and both of them have racked up substantial time on the ground in Ohio.