Set-orderedness as a generalization of kk-orderedness and cyclability

A graph GG is called kk-ordered if for any sequence of kk distinct vertices of GG, there exists a cycle in GG through these vertices in the given order. A vertex set SS is called cyclable in GG if there exists a cycle passing through all vertices of SS. We will define “set-orderedness” which is a natural generalization of kk-orderedness and cyclability. We also give a degree sum condition for graphs to satisfy “set-orderedness”. This is an extension of well-known sufficient conditions on kk-orderedness.