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.

Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for trianglefree graphs. * M.B., A.L. and J.N. are supported by ANR project GrR (ANR 18 CE40 0032). T. K. is supported by the grant no. 19-04113Y of the Czech Science Foundation (GAČR) and the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004), her visit to Bordeaux, where part of the research was conducted, was supported by Czech-French Mobility project 8J19FR027.1 Initially in the context of vertex colouring.Note that while some graphs need ∆(G) + 1 colours, some graphs can be edge-coloured with only ∆(G) colours. In the follow-up paper extending the result to multigraphs [Viz65], and later in a more publicly available survey paper [Viz68], Vizing asks whether an optimal colouring can always be reached through a series of Kempe changes, as follows.Question 1.3 ([Viz65]). For every simple graph G, for any integer k > χ ′ (G), for any k-edgecolouring α, is there a χ ′ (G)-edge-colouring that can be reached from α through a series of Kempe changes? Question 1.3 is in fact stated in the more general context of multigraphs. Note that neither Theorem 1.2 nor Question 1.3 implies that all colourings with fewer colours are reachable, i.e., there is no choice regarding the target colouring. We say two kedge-colourings are Kempe-equivalent if one can be reached from the other through a series of Kempe changes using colours from {1, . . . , k}. Question 1.3, if true, would imply [AC16] and the following conjecture of Mohar [Moh06], using the target χ ′ (G)-colouring as an intermediate colouring.Conjecture 1.4 ([Moh06]). For every simple graph G, all (∆(G) + 2)-edge-colourings are Kempe-equivalent.Mohar proved the weaker case where (χ ′ (G) + 2) colours are allowed.Theorem 1.5 ([Moh06]). For every simple graph G, all (χ ′ (G) + 2)-edge-colourings are Kempeequivalent.

In this paper, we study the dualization in distributive lattices, a generalization of the well known hypergraph dualization problem. We give a characterization of the complexity of the problem under various combined restrictions on graph classes and posets, including bipartite, split and co-bipartite graphs, and variants of neighborhood inclusion posets. In particular, we show that while the enumeration of minimal dominating sets is possible with linear delay in split graphs, the problem gets as hard as for general graphs in distributive lattices. More surprisingly, this result holds even when the poset coding the lattice is only comparing vertices of included neighborhoods in the graph. If both the poset and the graph are sufficiently restricted, we show that the dualization becomes tractable relying on existing algorithms from the literature.

No abstract

No abstract