Clique independence, domination and irredundance in graphs (Q2799813)

A set \(S\) of vertices in a graph \(G\) is a clique if the induced subgraph, denoted \(G[S]\), by \(S\) is a maximal complete subgraph of \(G\). The following concepts were introduced in this paper. A set \(S\) is called clique independent if the induced subgraph \(G[S]\) contains no clique in \(G\). A set \(S\) is a clique dominating set if every vertex \(v \in V- S\) is contained in a clique in the induced subgraph \(G[ S \cup \{v\}\). A set \(S\) is called clique irredundant if for every vertex \(v \in S\) there exists a vertex \(u \in V- S\) which is contained in a clique in \(G[ S \cup \{u\}]\) but is not contained in a clique in \(G[S - \{v\} \cup \{u\}]\). These concepts are generalizations of independence, domination, and irredundance in graphs. The authors obtained some results involving the introduced concepts for general graphs and triangle-free graphs.