Further results on outer independent 2-rainbow dominating functions of graphs

Abstract

Let G = (V(G), E(G)) be a graph. A function f : V(G) → ℙ({1, 2}) is a 2-rainbow dominating function if for every vertex v with f(v) = ∅, f(N(v)) = {1, 2}. An outer-independent 2-rainbow dominating function (OI2RD function) of G is a 2-rainbow dominating function f for which the set of all v 2208 V(G) with f(v) = ∅ is independent. The outer independent 2-rainbow domination number (OI2RD number) γoir2(G) is the minimum weight of an OI2RD function of G. In this paper, we first prove that n/2 is a lower bound on the OI2RD number of a connected claw-free graph of order n and characterize all such graphs for which the equality holds, solving an open problem given in an earlier paper. In addition, a study of this parameter for some graph products is carried out. In particular, we give a closed (resp. an exact) formula for the OI2RD number of rooted (resp. corona) product graphs and prove upper bounds on this parameter for the Cartesian product and direct product of two graphs.