Combinatorial characterization of ℓ-minimal graphs

Obtain a combinatorial characterization that determines exactly when a graph is [?] [?] [?] [?]

Background

An [?] [?] graph is a graph G satisfying r_+(G)=ℓ and |V(G)|=3ℓ. The paper proves that such graphs exist for every positive integer ℓ and constructs broad families of them, including stretched cliques in \hat{K}_{ℓ+2,ℓ-1} with clique number at most three. However, the paper also gives examples suggesting that membership in these stretched-clique families is not necessary. The unresolved question is therefore to identify a purely combinatorial criterion covering all ℓ-minimal graphs.