On a class of kernel-perfect and kernel-perfect-critical graphs
Let \(G=(V,E)\) be a directed graph. Then a set \(K\) of its vertices is called a kernel if \(K\) is an independent set, and \(K\cap \Gamma^{- 1}(K)=\varnothing\), \(K\cup\Gamma^{-1}(K)=V\), where \(\Gamma^{-1}(K)\) is the set of all the vertices \(x\) of \(G\) such that there exists \(y\in K\) with \((x,y)\) being an arc of \(G\). \(G\) is called a kernel-perfect graph if all induced subgraphs of \(G\), including \(G\), have a kernel, and it is called a kernel-perfect-critical graph if \(G\) does not have a kernel but every proper induced subgraph has. In the paper a new construction of both kernel-perfect graphs and kernel-perfect-critical graphs is presented.
scientific article; zbMATH DE number 417444