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A characterization of extremal graphs with no matching-cut

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 $G=(V,E)$, $|E|≥\lceil 3(|V|-1)/2\rceil$ , and constructed a large class of immune graphs that attain this lower bound for every value of $|V(G)|$, called $ABC$ graphs. They conjectured that every immune graph that attains this lower bound is an $ABC$ graph. We present a proof of this conjecture.
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Paul Bonsma. A characterization of extremal graphs with no matching-cut. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.135-138, ⟨10.46298/dmtcs.3398⟩. ⟨hal-01184354⟩

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