It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP. In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. Then we show using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent conclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one.

We prove a recent conjecture of Beisegel et al. that for every positive integer $k$, every graph containing an induced $P_k$ also contains an avoidable $P_k$. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for $k \in \{1,2\}$ (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvátal et al. 2002, which assumed cycle restrictions.  We provide a constructive and elementary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for K t -free graphs and variants. This answers a question of Kanté et al. about enumeration in bipartite graphs.

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this article we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for K t -free graphs and for several related graph classes. This answers a question of Kanté et al. about enumeration in bipartite graphs.