Dichotomies properties on computational complexity of S-packing coloring problems

TL;DR

This paper investigates the computational complexity of S-packing coloring problems, establishing dichotomies between NP-complete and polynomial-time solvable cases for specific graph classes and list sizes.

Contribution

It provides a comprehensive classification of the complexity of S-packing coloring problems for various list sizes and graph classes, including dichotomies for subcubic graphs.

Findings

Dichotomy between NP-complete and polynomial-time solvable cases for three-integer lists on subcubic graphs

Complexity classification for S-packing coloring with varying list sizes

Identification of polynomial and NP-complete instances based on list properties

Abstract

This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a non decreasing list of integers S = (s\_1 , ..., s\_k ), G is S-colorable, if its vertices can be partitioned into sets S\_i , i = 1,... , k, where each S\_i being a s\_i -packing (a set of vertices at pairwise distance greater than s\_i). For a list of three integers, a dichotomy between NP-complete problems and polynomial time solvable problems is determined for subcubic graphs. Moreover, for an unfixed size of list, the complexity of the S-packing coloring problem is determined for several instances of the problem. These properties are used in order to prove a dichotomy between NP-complete problems and polynomial time solvable problems for lists of at most four integers.