The generalized locally checkable problem in bounded treewidth graphs
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We introduce a new problem that generalizes some previous attempts of covering locally checkable problems under the same umbrella. Optimization and decision problems such as {k}-dominating set, b-coloring, acyclic coloring and connected dominating set, can be seen as instances of this new problem. We prove that this new problem can be solved, under mild conditions, in polynomial time for bounded treewidth graphs. As a consequence, we obtain polynomial-time algorithms to solve, for bounded treewidth graphs, Grundy domination and double Roman 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).
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