Locally checkable problems parameterized by clique-width

TL;DR

This paper develops a dynamic programming approach for solving locally checkable problems on graphs with bounded clique-width, extending to global properties and demonstrating fixed-parameter tractability for certain problems.

Contribution

It introduces a polynomial-time algorithm for 1-locally checkable problems on graphs of bounded clique-width and extends the framework to global properties, broadening solvable problem classes.

Findings

Polynomial-time algorithm for 1-locally checkable problems on bounded clique-width graphs.

Extension of the framework to global properties involving color class sizes.

Demonstration of fixed-parameter tractability for the $[k]-$Roman domination problem.

Abstract

We continue the study initiated by Bonomo-Braberman and Gonzalez in 2020 on $r$-locally checkable problems. We propose a dynamic programming algorithm that takes as input a graph with an associated clique-width expression and solves a $1$-locally checkable problem under certain restrictions. We show that it runs in polynomial time in graphs of bounded clique-width, when the number of colors of the locally checkable problem is fixed. Furthermore, we present a first extension of our framework to global properties by taking into account the sizes of the color classes, and consequently enlarge the set of problems solvable in polynomial time with our approach in graphs of bounded clique-width. As examples, we apply this setting to show that, when parameterized by clique-width, the $[k]-$Roman domination problem is FPT, and the $k$-community problem, Max PDS and other variants are XP.