Looking to solidify their unsettled situation at kicker, the Giants signed Randy Bullock, a former Houston Texans and Jets kicker, on Monday. That move could imperil the career of Josh Brown, who was suspended for one game by the N.F.L. last week for a 2015 arrest on a domestic violence charge.

Brown, a Giant for the last three years, was arrested by the police in Washington State over allegations involving his wife at the time. The charge against Brown was soon dropped, but Brown’s wife told the police that Brown had become physically violent with her on numerous occasions.

Aware of those statements, the N.F.L. conducted a 10-month investigation. It said that lacking the cooperation of Brown’s wife or law enforcement officials, it found “insufficient information to corroborate prior allegations.” The league also noted the lack of criminal charges.

Giants management has been silent on the subject for several days, although Coach Ben McAdoo expressed support of Brown shortly after the suspension was announced. Still, Bullock’s presence could present a difficult choice.