I am currently going through this. There is a thing that kids develop called "the poopie dance." Some kids have a painful movement and never want to go again so they hold it. Since they are afraid to go, they won't.
First thing is go the pharmacy and ask them for suppository for kids. I wouldn't do this often, maybe once or twice if needed, because it causes them to have more fear. Use benefiber when cooking or in your child's drink to add extra fiber. My doctor also had me use Pedilax, any drugstoe, and the worked wonderfully. Now that the Pedilax worked we have a "program" where we have to use Miralax and Benefiber for a month to make sure the movements stay soft and then wean him off the Miralax, Benefiber is a natural thing so it doesn't hurt to put in food. I would only do the suppository until you speak with your doctor, being that your child is only 1. There is a great book called Toddler 411 and they talk about just about every issue with toddler's and I actually read that before I saw the doctor and they told me to do the same thing I read in the book. I will say it is horrible until they get it out. My son would look and me, scream and cry, and I felt helpless. A warm bath is also good because it relaxes their bottom and may help them go.
I have been going through this on and off for a few months and just recently it became more often. Good luck