On Duality between Local Maximum Stable Sets of a Graph and its   Line-Graph
  G is a Koenig-Egervary graph provided alpha(G)+ mu(G)=|V(G)|, where mu(G) is the size of a maximum matching and alpha(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G if S is a maximum stable set of the closed neighborhood of S. Nemhauser and Trotter Jr. proved that any local maximum stable set is a subset of a maximum stable set of G. In this paper we demonstrate that if S is a local maximum stable set, the subgraph H induced by the closed neighborhood of S is a Koenig-Egervary graph, and M is a maximum matching in H, then M is a local maximum stable set in the line graph of G. 
Vadim E. Levit and Eugen Mandrescu