A note on the 2-rainbow bondage numbers in graphs
A 2-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of [Formula: see text]. The rainbow bondage number [Formula: see text] of a graph [Formula: see text] with maximum degree at least two is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. Dehgardi, Sheikholeslami and Volkmann, [The [Formula: see text]-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 [Formula: see text]-cell embedding on a surface with non-negative Euler characteristic.