On Well-\(k\)-Covered Graphs

A set \(S\) of vertices of a graph is \(k\)-independent if each vertex in \(S\) is adjacent to at most \(k-1\) other vertices in \(S\). A graph \(G\) is well-\(k\)-covered if every maximal \(k\)-independent set is maximum. We shall characterize the well-\(k\)-covered trees and for \(k=2\) all such graphs of girth \(8\) or more.