It is my understanding, per the State Dept. website, that unless the other parent is not on the passport at all, then you MAY be asked to provide additional documentation.
Obviously, if there is no second parent at all, the child's passport will reflect this (ie, there will only be one parent's name). If the other parent on the passport is deceased, you may be asked to provide a death certificate.
To obtain a passport for a minor requires the signatures of both parents (unless you submitted a court order explaining why two signatures could not be obtained at the time you applied for the passport). Even if you have full legal custody, the court documents MAY not be sufficient. This is in part because states can have different definitions of sole custody. I was awarded joint legal custody and sole physical custody. My ex has visitation rights (which he would have been awarded even if I had received full legal custody), so we are traveling to Germany with a letter just to be safe.
In fact, the state dept recommends that you travel with the letter even if both parents are still married and one just isn't traveling. IMO, it's not worth the risk of being detained and missing your trip over this issue.