Roman and inverse Roman domination in graphs

Motivated by the article in Scientific American [8], Michael A Henning and Stephen T. Hedetniemi explored the strategy of defending the Roman Empire. Cockayne defined Roman dominating function (RDF) on a Graph G = (V, E) to be a function f : V -> {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. For a real valued function f : V -> R the weight of f is w (f) = Sigma(v is an element of V) f (v). The Roman Domination Number (RDN) denoted by gamma(R)(G) is the minimum weight among all RDF in G. If V - D contains a Roman dominating function f(1) : V -> { 0, 1, 2}, where D is the set of vertices v for which f (v) > 0. Then f(1) is called inverse Roman dominating function (IRDF) on a graph G w.r.t. f. The inverse Roman domination number (IRDN) denoted by gamma(1)(R)(G) is the minimum weight among all IRDF in G. In this paper we find few results of RDN and IRDN.