# Maximal Unit interval graph construction

Is it possible to construct a unit interval graph with $n$ vertices and clique number $k$ such that any other unit interval graph with same number of vertices and clique number is a sub-graph of that graph or isomorphic to a sub-graph of that graph?

• The graph is simply the $(k-1)$th power of the path on $n$ vertices, $P_n^{k-1}$. – Andrew D. King Jun 17 '14 at 23:28