On \(1\)-\(Z_m\)-Well-Covered and Strongly \(Z_m\)-Well-Covered Graphs

A graph \(G\) is \(Z_m\)-well-covered if \(|I| \equiv |J| \pmod{m}\), for all \(I\), \(J\) maximal independent sets in \(V(G)\). A graph \(G\) is a \(1-Z_m\)-well-covered graph if \(G\) is \(Z_m\)-well-covered and \(G\setminus\{v\}\) is \(Z_m\)-well-covered, \(\forall v \in V(G)\). A graph \(G\) is strongly \(Z_m\)-well-covered if \(G\) is a \(Z_m\)-well-covered graph and \(G\setminus\{e\}\) is \(Z_m\)-well-covered, \(\forall e \in E(G)\). Here we prove some results about \(1-Z_m\)-well-covered and strongly \(Z_m\)-well-covered graphs.