Gingerbread houses in January

I made gingerbread houses with the kids over Christmas break. I cheated and used graham crackers, which ended up taking longer than just making the dough and baking the pieces because I found out that graham crackers don't cut well, Anyway, as any good mom, I told my kids that they couldn't eat the candy off the houses because the frosting is old and hard and makes a crumbly mess.

So, of course, they've been sneaking pieces since I put their houses up on the piano as decorations. Can I throw them out now or is there an allotted time frame before they can be tossed in the great abyss that is our city issued garbage can?

It is January.