3 research outputs found
Thinness of product graphs
The thinness of a graph is a width parameter that generalizes some properties
of interval graphs, which are exactly the graphs of thinness one. Many
NP-complete problems can be solved in polynomial time for graphs with bounded
thinness, given a suitable representation of the graph. In this paper we study
the thinness and its variations of graph products. We show that the thinness
behaves "well" in general for products, in the sense that for most of the graph
products defined in the literature, the thinness of the product of two graphs
is bounded by a function (typically product or sum) of their thinness, or of
the thinness of one of them and the size of the other. We also show for some
cases the non-existence of such a function.Comment: 45 page
On -stable locally checkable problems parameterized by mim-width
In this paper we continue the study of locally checkable problems under the
framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on
graphs of bounded mim-width. We study which restrictions on a locally checkable
problem are necessary in order to be able to solve it efficiently on graphs of
bounded mim-width. To this end, we introduce the concept of -stability of a
check function. The related locally checkable problems contain large classes of
problems, among which we can mention, for example, LCVP problems. We give an
algorithm showing that these problems are XP when parameterized by the
mim-width of a given binary decomposition tree of the input graph, that is,
that they can be solved in polynomial time given a binary decomposition tree of
bounded mim-width. We explore the relation between -stable locally checkable
problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke,
2022), and show that both frameworks model the same family of problems. We
include a list of concrete examples of -stable locally checkable problems
whose complexity on graphs of bounded mim-width was open so far