Minimal dominating sets enumeration with FPT-delay parameterized by the degeneracy and maximum degree

Overview & Synopsis

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$, for an appropriate definition of degeneracy. Recently at IWOCA 2019, Conte, Kanté, Marino, and Uno asked whether, even for a more restrictive notion of degeneracy, this XP-delay algorithm parameterized by $d$ could be made FPT-delay parameterized by $d$ and the maximum degree $\Delta$, i.e., an algorithm with delay $f(d, \Delta) \cdot n^{O(1)}$ for some computable function $f$. We answer this question in the affirmative whenever the hypergraph corresponds to the closed neighborhoods of a graph, i.e., we show that the intimately related problem of enumerating minimal dominating sets in graphs admits an FPT-delay algorithm parameterized by the degeneracy and the maximum degree.