On maximal Roman domination in graphs

A Roman dominating function (RDF) for a graph is a function satisfying the condition that every vertex u of G for which is adjacent to at least one vertex v of G for which . The weight of a RDF f is the sum , and the minimum weight of a RDF for G is the Roman domination number, of G. A maximal RDF for a graph G is a RDF f such that is not a dominating set of G. The maximal Roman domination number, of a graph G equals the minimum weight of a maximal RDF for G. We first show that determining the number for an arbitrary graph G is NP-complete even when restricted to bipartite or planar graphs. Then, we characterize connected graphs G such that . We also provide a characterization of all trees T of order n such that .