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.

翻译：图的$b$-着色是一种正常顶点着色，其中每个颜色类包含一个顶点，该顶点在其邻域中能看到所有其他颜色。$b$-着色问题的任务是判断图是否允许使用$k$种颜色进行$b$-着色，该问题在一般情况下是NP完全的，但在树上可在多项式时间内求解。此外，已知当以树宽为参数时，$b$-着色属于XP类，但对所有$t \\in \\mathbb{N}$都是W[$t$]-难的。事实上，仅有极少参数（如顶点覆盖数）已知对$b$-着色允许FPT算法。本文考虑一个比树宽更严格、衡量与树相似性的参数——反馈边数，并证明$b$-着色在该参数化下是固定参数可处理的。我们的算法结合了参数化算法学中的标准技术与针对树的$b$-着色多项式时间算法中的问题特定思路。此外，我们提出了一个以到共簇距离为参数的$b$-着色FPT算法，该参数衡量与完全多部图的相似性。最后，基于已知结果我们提出若干观察，包括当以树深为参数时$b$-着色是W[$1$]-难的。