Saturation of Ordered Graphs

TL;DR

This paper explores the saturation problem in ordered and cyclically ordered graphs, establishing a dichotomy where the saturation function is either bounded or linear, and provides classifications and examples for these behaviors.

Contribution

It extends the saturation problem to ordered and cyclically ordered graphs, proving a dichotomy and analyzing specific subclasses like linked matchings.

Findings

Saturation function is either bounded or linear for these graphs.

Many linked matchings with up to three links have bounded saturation functions.

Complete characterization of graphs with bounded semisaturation functions.

Abstract

Recently, the saturation problem of $0$-$1$ matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of P\'alv\"olgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.