A new approach on locally checkable problems

TL;DR

This paper introduces a new framework for locally checkable problems on bounded treewidth graphs, enabling polynomial-time solutions for several previously unsolved problems and extending the class of solvable problems through graph power analysis.

Contribution

The paper presents a novel framework that extends previous results, allowing polynomial-time solutions for new problems on bounded treewidth graphs and their powers.

Findings

Polynomial-time algorithms for double Roman and Grundy domination.

Extension of polynomial-time solvability to graph powers with bounded degree and treewidth.

Broader class of problems, including distance coloring and domination, solvable in polynomial time.

Abstract

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).