On the 2-rainbow bondage number of planar graphs

A 2-rainbow dominating function (2RDF) on a graph G = (V, E) is a function f from the vertex set V to the set of all subsets of the set {1, 2} such that for any vertex v is an element of V with f (v) = empty set the condition boolean OR(u is an element of N(v)) f(u) = {1, 2} is fulfilled. The weight of a 2RDF f is the value omega(f) = Sigma(v is an element of v(G)) vertical bar f(v)vertical bar. The 2-rainbow domination number, denoted by gamma(r2) (G), is the minimum weight of a 2RDF on G. The rainbow bondage number b(r2)(G) of a graph G with maximum degree at least two, is the minimum cardinality of all sets E' subset of E(G) for which gamma(r2)(G - E') > gamma(r2)(G). Dehgardi, Sheikholeslami and Volkmann, [Discrete Appl. Math. 174 (2014), 133-139] proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper we improve this result.