The news of Rep. Gabrielle Giffords’ (D-Ariz.) ensuing resignation is still just a day old, but numerous Democratic names have been floated as potential candidates for her seat.

One name that can be crossed off the list is Homeland Security Secretary Janet Napolitano, a former two-term governor of Arizona. Few if anyone believed she would leave her job for a seat in the House, but her candidacy likely would have cleared the field.

“Secretary Napolitano is honored to serve the president and the American people as secretary of homeland security,” department spokesman Matt Chandler said. “While she has no intention of running for Congresswoman Giffords’ seat, she thanks Congresswoman Giffords for her service to Arizona and wishes her the very best as she continues her remarkable recovery.”

Both parties view the special election to replace Giffords as competitive. The process won’t begin until she makes her resignation official later this week.