On partial cubes, well-graded families and their duals with some applications in graphs

TL;DR

This paper explores the duality of well-graded, extremal, and maximum set systems, characterizes graphs with specific neighborhood properties, and relates these to VC-dimension and half-graphs, advancing understanding in combinatorics and graph theory.

Contribution

It introduces a duality framework for special set systems and applies it to characterize graphs with particular neighborhood structures, linking to VC-dimension and half-graphs.

Findings

Characterization of dual well-graded, extremal, and maximum set systems.

Identification of graph classes with neighborhood systems that are well-graded or extremal.

Connection between these graph classes and VC-dimension, half-graphs.

Abstract

Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the context of these classes of set systems and then use the obtained results for studying graphs. More specifically, we are concerned with the characterization of the finite set systems which themselves and their dual systems are both well-graded, extremal or maximum. On the way to this goal, and maybe also of independent interest, we study the structure of the well-graded families with the property that the size of the system is not much bigger than the size of its essential domain, that is, the set of elements of the domain which are shattered by the system as single element subsets. As another target of the paper, we use the above results to characterize graphs whose set systems of open or closed neighbourhoods, cliques or independent sets are well-graded, extremal or maximum. We clarify the relation of such graphs to the celebrated half-graphs. Through the paper, we frequently relate our investigations to the VC-dimension of the systems. Also we use one-inclusion graphs associated to set systems as an important technical tool.