Finding $b$-colorings Using Feedback Edges

TL;DR

This paper proves that finding $b$-colorings is fixed-parameter tractable when parameterized by feedback edge number, using a combination of standard and problem-specific techniques, and also explores related parameters.

Contribution

It introduces an FPT algorithm for $b$-coloring based on feedback edge number and extends the analysis to distance to co-cluster, advancing understanding of parameterized complexity.

Findings

FPT algorithm for $b$-coloring with feedback edge number

FPT algorithm for $b$-coloring with distance to co-cluster

W[$1$]-hardness when parameterized by tree-depth

Abstract

A $b$-coloring of a graph is a proper vertex coloring such that each color class contains a vertex that sees all other colors in its neighborhood. The $b$-coloring problem, in which the task is to decide whether a graph admits a $b$-coloring with $k$ colors, is NP-complete in general but polytime solvable on trees. Moreover, it is known that $b$-coloring is in XP but W[$t$]-hard for all $t \in \mathbb{N}$ when parameterized by tree-width. In fact, only very few parameters, such as the vertex cover number, were known to admit an FPT algorithm for $b$-coloring. In this paper, we consider a more restrictive parameter measuring similarity to trees than tree-width, namely the feedback edge number, and show that $b$-coloring is fixed-parameter tractable under this parameterization. Our algorithm combines standard techniques used in parameterized algorithmics with the problem-specific ideas used in the polytime algorithm for trees. In addition, we present an FPT algorithm for $b$-coloring parameterized by distance to co-cluster, which is a parameter measuring similarity to complete multipartite graphs. Finally, we make several observations based on known results, including that $b$-coloring is W[$1$]-hard when parameterized by tree-depth.