On the Helly Subclasses of Interval Bigraphs and Circular Arc Bigraphs

Abstract

A bipartite graph \(G=(U,V,E)\) is an interval bigraph if and only if there is a one to one correspondence between \(U \cup V\) and a family of intervals on the number line such that two vertices of opposing partite sets are neighbors precisely if their corresponding intervals intersect. Interval bigraphs, as well as many subclasses, have been extensively studied by multiple researchers along the years, and many results on their structural and computational properties have been discovered. A bipartite graph \(G=(U,V,E)\) is a circular arc bigraph if and only if there is a one to one correspondence between \(U \cup V\) and a family of arcs on a circle such that two vertices of opposing partite sets are neighbors precisely if their corresponding arcs intersect. While it is a generalization of interval bigraphs, it remains a relatively unexplored topic. Few studies about the class and its proper, unit and Helly subclasses have been presented. In this work, we study some subclasses of these classes. We provide forbidden structure characterizations for the Helly subclass of interval bigraphs, as well as the class of non-bichordal Helly circular arc bigraphs. We also prove that Helly interval bigraphs are a subclass of proper interval bigraphs, and that non-bichordal Helly circular arc bigraphs are a subclass of proper circular arc bigraphs.