Packing functions and dominating functions of countable graphs

We define for countable graphs, parameters called closed packing index, open packing index, domination index and total domination index; these four are extensions of respectively fractional closed packing number, fractional open packing number, fractional domination number and fractional total domination number for finite graphs. We compute the closed packing index (resp. the open packing index) of the join of two disjoint countable graphs G and H in terms of the corresponding parametric values for G and H and for any real α⩾1, we construct a connected graph for which, each of the above described four indexes is equal to α.