Abstract
Given a proper coloring f of G, a vertex u is a b-vertex if it is adjacent to every color class distinct from its own. It is said to be a b-coloring if each color class contains at least one b-vertex, and a fall coloring if all vertices are b-vertices. Also, if f is a fall coloring of an induced subgraph H of G, then we say that f is a subfall coloring of G. In this paper, we provide algorithms for each of the decision problems related to these colorings whose running times are FPT when parameterized by the number of colors plus the treewidth of the input graph.
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