On the Roman Bondage Number of Planar Graphs

A Roman dominating function on a graph G is a function f : V(G) -> {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value f (V(G)) = Sigma(u is an element of V(G)) f (u). The Roman domination number, gamma(R)(G), of G is the minimum weight of a Roman dominating function on G. The Roman bondage number b(R)(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(R)(G - E') > gamma(R)(G). In this paper we present different bounds on the Roman bondage number of planar graphs.