Minimum fractional dominating functions and maximum fractional packing functions

The fractional analogues of domination and packing in a graph form an interesting pair of dual linear programs in that the feasible vectors for both LPs have interpretations as functions from the vertices of the graph to the unit interval; efficient (fractional) domination is accomplished when a function simultaneously solves both LPs. We investigate some structural properties of the functions thus defined and classify some families of graphs according to how and whether the sets of functions intersect. The tools that we develop have proven to be useful in approaching problems in fractional domination and in the broader theory of domination.