On Duality between Local Maximum Stable Sets of a Graph and Its Line-Graph

G is a König-Egerváry graph provided α(G) +μ(G) = |V (G)|, where μ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set, [3], [22]. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪N(S), where N(S) is the neighborhood of S, [12]. Nemhauser and Trotter Jr. proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G, [20]. In this paper we demonstrate that if S ∈ Ψ(G), the subgraph H induced by S∪N(S) is a König-Egerváry graph, and M is a maximum matching in H , then M is a local maximum stable set in the line graph of G.