  By providing a new framework, we extend previous results on locally checkable
problems in bounded treewidth graphs. As a consequence, we show how to solve,
in polynomial time for bounded treewidth graphs, double Roman domination and
Grundy domination, among other problems for which no such algorithm was
previously known. Moreover, by proving that fixed powers of bounded degree and
bounded treewidth graphs are also bounded degree and bounded treewidth graphs,
we can enlarge the family of problems that can be solved in polynomial time for
these graph classes, including distance coloring problems and distance
domination problems (for bounded distances).
