Saturation Spectrum of Paths and Stars

A graph is -saturated if is not a subgraph of but the addition of any edge from to results in a copy of . The minimum size of an -saturated graph on vertices is denoted sat( ), while the maximum size is the well studied extremal number, ex( ). The saturation spectrum for a graph is the set of sizes of saturated graphs between sat( ) and ex( ). In this paper we completely determine the saturation spectrum of stars and we show the saturation spectrum of paths is continuous from sat( ) to within a constant of ex( ) when is sufficiently large.