DOMINATION, FRACTIONAL DOMINATION, 2-PACKING, AND GRAPH PRODUCTS

Let P2(G), gamma(f)(G), and gamma(G) be the 2-packing number, fractional domination number, and domination number, respectively, of a graph G. Domke, Hedetniemi, and Laskar [Congress. Numer., 66 (1989), pp. 227-238] showed that P2(G) less-than-or-equal-to gamma(f)(G) less-than-or-equal-to gamma(G). Examples are given with P2(G) < gamma(f)(G) = gamma(G) and P2(G) = gamma(f)(G) < gamma(G). Let G + H and G . H be the Cartesian product and strong direct product, respectively, of graphs G and H. For all G and H, it is shown that P2(G)P2(H) less-than-or-equal-to P2(G . H) less-than-or-equal-to P2(G)gamma(f)(H) and gamma(G)gamma(f)(H) less-than-or-equal-to gamma(G . H) less-than-or-equal-to gamma(G)gamma(H). These relations are also independent. Relations involving P2(G + H), gamma(f)(G + H), and gamma(G + H) are examined. An unresolved issue involves a conjecture of Vizing: For all G and H, is gamma(G + H) greater-than-or-equal-to gamma(G)gamma(H)?