On the \(2\)-Rainbow Bondage Number of Planar Graphs

Abstract

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 \(v \in V\) with \(f(v) = \emptyset\) the condition \(\cup_{u \in N(v)} f(u) = \{1, 2\}\) is fulfilled. The weight of a 2RDF \(f\) is the value \(w(f) = \sum_{v \in V(G)} |f(v)|\). 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 \(E’ \subseteq E(G)\) 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.