(Independent) Roman 2-Domination Number on Graphs

For a graph Formula: see text, a function Formula: see text is a Roman 2-dominating function (R2DF) if Formula: see text satisfies the following condition: if Formula: see text then there is Formula: see text such that Formula: see text or there is Formula: see text such that Formula: see text. Let Formula: see text denote the set of vertices assigned Formula: see text by function Formula: see text. The weight of a R2DF Formula: see text is the sum Formula: see text. The Roman 2-domination number is the minimum weight of a R2DF on Formula: see text, denoted by Formula: see text. A function Formula: see text is an independent Roman 2-dominating function (IR2DF) if Formula: see text is a R2DF and Formula: see text is an independent set. In this paper, we initially study the influence of edge addition on the Roman 2-domination number, and for two positive integer Formula: see text and Formula: see text, we construct a graph Formula: see text and an induced subgraph Formula: see text of Formula: see text such that Formula: see text and Formula: see text. Based on this construction, we conclude that there is no relation between the Roman 2-domination number of a graph and its induced subgraph. Furthermore, we give a bound for the independent Roman Formula: see text-domination number on proper interval graphs, and a linear-time algorithm for computing the independent Roman Formula: see text-domination number of unicyclic graphs with restrictions.