Some operations of graphs that preserve the property of well-covered by monochromatic paths

A graph is called well-covered if every maximal independent set of vertices of G is a maximum independent set; recall that S is independent if no two of its vertices are adjacent. In this paper we define the concept of well-covered by monochromatic paths graphs which is a variation of well-covered graphs. We consider some classical constructions of graphs: G-join of graphs and duplication of a subset of vertices. We also give necessary and sufficient conditions for well-coveredness by monochromatic paths of these graphs.