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For each vertex \(s\) of the subset \(S\) of vertices of a graph \(G\), we define Boolean variables \(p\), \(q\), \(r\) which measure the existence of three kinds of \(S\)-private neighbors (\(S\)-pns) of \(s\). A \(3\)-variable Boolean function \(f = f(p, q, r)\) may be considered as a compound existence property of \(S\)-pns. The set \(S\) is called an \(f\)-set of \(G\) if \(f = 1\) for all \(s \in S\), and the class of all \(f\)-sets of \(G\) is denoted by \(\Omega_f\). Special cases of \(\Omega_f\) include the
independent sets, irredundant sets, open irredundant sets, and CO-irredundant sets of \(G\).
There are \(62\) non-trivial families \(\Omega_f\) which include the \(7\) families of a framework proposed earlier by Fellows, Fricke, Hedetniemi, and Jacobs. The functions \(f\) for which \(\Omega_f\) is hereditary for any graph \(G\) are determined. Additionally, the existence and properties of \(f\)-Ramsey numbers (analogues of the elusive classical Ramsey numbers) are investigated, and future directions for the theory of the classes \(\Omega_f\) are considered.