List homomorphisms of graphs with bounded degrees

In a series of papers, we have classified the complexity of list homomorphism problems. Here, we investigate the effect of restricting the degrees of the input graphs. It turns out that the complexity does not change (except when the degree bound is two). We obtain similar results on restricting the size of the lists.We contrast these results with facts about variants of the list homomorphism problem, where restricting the degrees can have an important effect on the complexity.