Kernels in the closure of coloured digraphs (Q2725188)

The closure of an arc coloured digraph \(D=(V,A)\) is defined as the coloured multidigraph \(\zeta(D)=(V,B),\) where \(B=\bigcup_i\{(u,v)\) with the colour \(i\) : there exists a monochromatic path of colour \(i\) from \(u\) to \(v\) in \(D\)\}. Let \(C_3\) denote the class of all 3-coloured (i.e. coloured with 3 different colours) directed cycles of order~3 and let \(T_3\) denote the class of all 3-coloured transitive tournaments of order 3. The authors prove that if \(D\) is obtained from an arc coloured tournament by deleting one arc and \(D\) is \(\{C_3\cup T_3\}\)-free, then every induced subdigraph of its closure \(\zeta(D)\) has a kernel (i.e. independent and absorbent set of vertices).