Combinatorial characterization of LS-plus-minimal graphs

Characterize exactly when a given graph is -minimal, and thereby obtain a combinatorial characterization of all graphs whose LovszSchrijver operator rank equals and whose order is .

Background

The paper proves that for every positive integer , graphs with LovszSchrijver rank and exactly vertices exist, calling such graphs -minimal. It gives broad families of examples, including stretched cliques in with clique number at most three, and establishes that the class is substantially richer than the initially studied stretched-clique constructions.

The authors note that membership in is sufficient but not necessary for -minimality, and they present computational evidence of additional -minimal graphs outside this family. A complete structural description of all -minimal graphs therefore remains unresolved.