I'm working with graphs $G$ that have the following property: there is an ordering $v_1, \ldots, v_n$ of the vertices of $G$ such that the neighbours of $v_k$ among $\{v_1, \ldots, v_{k-1}\}$ induce a (possibly empty) clique. Examples include trees, indifference graphs, and quasi-threshold graphs.

**Question**: Is there a name for the class of graphs that satisfy this property?

[What I'm really using about the graphs is that there is an ordering of the vertices such that if the graph is revealed vertex-by-vertex in that order, then the sequence of (partial) chromatic polynomials forms a chain: earlier chromaric polynomials are divisors of later ones.]