A characterization of extremal graphs without matching-cuts

Abstract

A graph is called (matching-)immune if it has no edge cut that is also a matching. Farley and Proskurowski proved that for all immune graphs $G=(V,E)$, $|E|\geq \lceil 3(|V|-1)/2\rceil$, and constructed a large class of immune graphs attaining this lower bound for every value of $|V(G)|$, called ABC graphs. In this paper, we prove their conjecture that every immune graph that attains this lower bound is an ABC graph