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Abstract
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A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex
set V (G) to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V (G) with
f(v) = ∅, the condition Su∈N(v) f(u) = {1, 2} is fulfilled. The weight of a 2RDF f is the
value ω(f) = Pv∈V (G) |f(v)|. The 2-rainbow domination number of a graph G, denoted
by γr2(G), is the minimum weight of a 2RDF of G. The rainbow bondage number br2(G)
of a graph G with maximum degree at least two is the minimum cardinality of all sets
E ⊆ E(G) for which γr2(G − E) > γr2(G). Dehgardi, Sheikholeslami and Volkmann,
[The k-rainbow bondage number of a graph, 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 generalize their result for graphs which admit a 2-cell embedding on a surface
with non-negative Euler characteristic.
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