BOUNDS RELATING GENERALIZED DOMINATION PARAMETERS

The domination number gamma(G) and the total domination number gamma(t)(G) of a graph G are generalized to the K(n)-domination number gamma(Kn)(G) and the total K(n)-domination number gamma(Kn)t(G) for n greater-than-or-equal-to 2, where gamma(G) = gamma(K2)(G) and gamma(t)(G) = gamma(K2)(G). K(n)-connectivity is defined and, for every integer n greater-than-or-equal-to 2, the existence of a K(n)-connected graph G of order at least n + 1 for which gamma(Kn)(G) + gamma(Kn)t(G) = ((3n - 2)/n2)p(G) is established. We conjecture that, if G is a K(n)-connected graph of order at least n + 1, then gamma(Kn)(G) + gamma(Kn)t(G) less-than-or-equal-to ((3n - 2)/n2)p(G). This conjecture generalizes the result for n = 2 of Allan, Laskar and Hedetniemi. We prove the conjecture for n = 3. Further, it is shown that if G is a K 3-connected graph of order at least 4 that satisfies the condition that, for each edge e of G, G - e contains at least one K3-isolated vertex, then gamma(K)3(G) + gamma(K3)t(G) less-than-or-equal-to (3p)/4 and we show that this bound is best possible.