Each king on an n×n chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the king. A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set. We prove that the maximum size of an irredundant set of kings is bounded between (n− 1)=3 and n=3, and that the minimum size of a ...

For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s, S), q = q(s, S) and r = r(s, S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p, q, r) may be considered as a compound existence property of S-pns. The subset S is called an f -set of G if f = 1 for all s ∈ S and the class of f -se...

Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗ and parameter p pairs are multiplicative, that is, p(G ⊗ H) ≥ p(G)p(H) for all graphs G and H or p(G⊗H) ≤ p(G)p(H) for all graphs G and H. The parameters are related to independence, domination and ir...

The relationship ρL(G) ≤ ρ(G) ≤ γ (G) between the lower packing number ρL(G), the packing number ρ(G) and the domination number γ (G) of a graph G is well known. In this paper we establish best possible bounds on the ratios of the packing numbers of any (connected) graph to its six domination-related parameters (the lower and upper irredundance numbers ir and IR, the lower and upper independenc...