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

A graph G is called k-ordered if for any sequence of k distinct vertices of G, there exists a cycle in G through these vertices in the order. A vertex set S is called cyclable in G if there exists a cycle passing through all vertices of S. We will define “set-orderedness” which is a natural generalization of k-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 k-orderedness.