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“For Hickenlooper, this is a bad thing,” said Denver pollster Floyd Ciruli. “As a moderate governor, he’s much better off having a check and balance that acts as a moderating influence on the legislature.”

Not every state requires a supermajority to override a gubernatorial veto. In some states, a veto may be overcome with a simple majority of the legislature, while in other states, an override may require two-thirds, three-fourths or three-fifths votes of both chambers.

Other executives on the hot seat are the three governors facing veto-proof majorities from another party. Both houses of the Arkansas General Assembly flipped from Democratic to Republican control on Election Day. Democrat Mike Beebe is the governor of the state, which requires only a simple majority for an override.

Another Democrat, Missouri Gov. Jay Nixon, will continue to deal with veto-proof Republican legislative majorities. Rhode Island Gov. Lincoln Chafee, an independent, also will have to wield a veto that one party has the votes to override, although the former Republican senator supported President Obama’s re-election and sides with Democrats on most issues.

One-party rule was common in the 1960s and 1970s, but for different reasons. Prior to the 1980s, Democrats dominated state legislatures in the South and regularly held veto-proof majorities, while Republicans rarely held such control outside states such as Utah.

As of the latest tally, Republicans account for 16 of the 25 states with veto-proof majorities, while Democrats represent nine. Many of those Republican states are in the South, which has all but completed its full political-party makeover.

“The difference today is that most of the states with two-thirds majorities are Republican,” Mr. Kurtz said. “And that’s never happened before.”