A main question in graphical models and causal inference is whether, given a probability distribution P (which is usually an underlying distribution of data), there is a graph (or graphs) to which P is faithful. The main goal of this paper is to provide a theoretical answer to this problem. We work with general independence models, which contain probabilistic independence models as a special ca...

We consider the evaluation of first-order queries over classes of databases that are nowhere dense. The notion of nowhere dense classes was introduced by Nešetřil and Ossona de Mendez as a formalization of classes of “sparse” graphs and generalizes many well-known classes of graphs, such as classes of bounded degree, bounded treewidth, or bounded expansion. It has recently been shown by Grohe, ...

We give the rst algorithmic result using the bidi-rected cut relaxation for the metric Steiner tree problem: a 3=2+ factor approximation algorithm, for any > 0, for the special case of quasi-bipartite graphs; these are graphs that do not have edges connecting pairs of Steiner vertices. This relaxation is conjectured to have integrality gap close to 1; the worst example known has integrality gap...

The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. Our goal is to demonstrate that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, ...

Jeager et al introduced a concept of group connectivity as an generalization of nowhere zero flows and its dual concept group coloring, and conjectured that every 5-edge connected graph is Z3-connected. For planar graphs, this is equivalent to that every planar graph with girth at least 5 must have group chromatic number at most 3. In this paper we show that if G is a plane graph with girth at ...