Limits of Structures and the Example of Tree-Semilattices

The notion of left convergent sequences of graphs introduced by Lov\' asz et al. (in relation with homomorphism densities for fixed patterns and Szemer\'edi's regularity lemma) got increasingly studied over the past $10$ years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general framework for convergence of sequences of structures. In particular, the authors introduced the notion of $QF$-convergence, which is a natural generalization of left-convergence. In this paper, we initiate study of $QF$-convergence for structures with functional symbols by focusing on the particular case of tree semi-lattices. We fully characterize the limit objects and give an application to the study of left convergence of $m$-partite cographs, a generalization of cographs

Tree-depth and vertex-minors

Abstract In a recent paper Kwon and Oum (2014), Kwon and Oum claim that every graph of bounded rank-width is a pivot-minor of a graph of bounded tree-width (while the converse has been known true already before). We study the analogous questions for “depth” parameters of graphs, namely for the tree-depth and related new shrub-depth. We show how a suitable adaptation of known results implies that shrub-depth is monotone under taking vertex-minors, and we prove that every graph class of bounded shrub-depth can be obtained via vertex-minors of graphs of bounded tree-depth. While we exhibit an example that pivot-minors are generally not sufficient (unlike Kwon and Oum (2014)) in the latter statement, we then prove that the bipartite graphs in every class of bounded shrub-depth can be obtained as pivot-minors of graphs of bounded tree-depth