If a parent is available to care for the child, what can be done if the child is left home alone?

Call your lawyer or contact whatever your states version of Domestic Relations is. They are located usually in the county court house listings. Explain the situation to them and see what they advise.

I think 13 is too young to be alone all day especially if she is uncomfortable with it.

Have you asked if you could pick her up and return her so it will be convenient for him? Having been a single parent and having to transport my son for childcare in the past, I can understand that he might not want to add an hour or more (mine was two hours per day) commute to an already 10 hour day. My son treasured the day he turned 12 years old and was allowed to stay home alone and to be honest I did too. I work weird hours and carrying a sleeping 50 lb 6 year old at 11 PM up a few flights of stairs was not fun. Maybe if it will make his life easier your ex would go along and you wouldn't have to seek legal means.

Or do you and your ex not get along in general and he's doing this to get back at you? Legally, this is a gray area. There are no laws determining when a child can be left home alone. If you call DHS, they will consider the child's feelings of whether s/he is comfortable, scared or lonely, how long s/he will home alone, whether the child knows what to do in an emergency, and if adequate food and shelter is being provided. The biggest criteria is if there is a problem, is the child playing with fire, running the streets, being a public nuisance or endangering his or her own safety? The problem is that DHS is horribly underfunded and they require there to be a pretty good problem before they intervene, if they do even then.

Good luck!

Depending on how long she would be alone for, there is a limited case to be made that a 13 year old does not have the necessary independence to handle that for an extended period of time.

However, the custody agreement is binding, so you would either need to jointly agree to modify the arrangement, or petition the court to change the arrangement. Obviously the former, if possible, will be a lot less expensive and painful.

Perhaps a time trade, weekdays for weekends might be acceptable?