This fall, as Democrats and Republicans vie for control of state legislatures across the country, much of the attention has focused on states like Michigan, Pennsylvania and Arizona, where one or two seats could tip the balance.

But in a handful of states, the legislature is dominated by one party, while the governor’s office is held by another. In those states, a high-stakes effort is underway to either preserve an existing supermajority — which confers the ability to override a governor’s veto — or to break it.

Nowhere has the battle been more magnified than in North Carolina, a heavily gerrymandered state where Republicans hold the 60 percent minimum required for an override in both chambers, even though registered Democratic voters outnumber Republicans.

Last year, after State Representative Tricia Cotham unexpectedly switched her party affiliation from Democrat to Republican, Republican leaders were able to enact a 12-week limit on most abortions, overriding Gov. Roy Cooper, a Democrat.

Now, Republican legislators cannot afford to lose a single seat, in either chamber, if they want to continue to override his vetoes.

“Remember, one legislator delivered the supermajority,” Beth Helfrich, a Democrat running for the state House of Representatives, said during a recent town hall in Davidson, N.C. “My math says one can break it.”