(Open) packing number of some graph products

The packing number of a graph G is the maximum number of closed neighborhoods of vertices in G with pairwise empty intersections. Similarly, the open packing number of G is the maximum number of open neighborhoods in C with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters

Global Domination and Packing Numbers

For a graph G = (V, E), X subset of V is a global dominating set if X dominates both G and the complement graph (G) over bar. A set X C V is a packing if its pairwise members are distance at least 3 apart. The minimum number of vertices in any global dominating set is gamma(g)(G), and the maximum number in any packing is p(G). We establish relationships between these and other graphical invariants, and characterize graphs for which p(G) = p(G). Except for the two self complementary graphs on 5 vertices and when G or (G) over bar has isolated vertices, we show gamma(g)(G) \u3c = left perpendicular n/2 right perpendicular, where n = vertical bar V vertical bar

Global Domination And Packing Numbers

For a graph G = (V, E), X ⊆ V is a global dominating set if X dominates both G and the complement graph G. A set X ⊆ V is a packing if its pairwise members are distance at least 3 apart. The minimum number of vertices in any global dominating set is γ8(G), and the maximum number in any packing is ρ(G). We establish relationships between these and other graphical invariants, and characterize graphs for which p(G) = ρ(G). Except for the two self-complementary graphs on 5 vertices and when G or ̄ has isolated vertices, we show γg(G) ≤⌊n/2⌋, where n = |V|