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Letbe an integer and  G  a simple graph with vertex set  V ( G ). Let  f  be a function that assigns labels from the setto the vertices of  G . For a vertexthe active neighbourhood  AN ( v ) of  v  is the set of all vertices  w  adjacent to  v  such thatA [ k ]-Roman dominating function (or [ k ]-RDF for short) is a functionsatisfying the condition that for any vertexwith  f ( v ) &lt;  k ,The weight of a [ k ]-RDF isand the [ k ]-Roman domination numberof  G  is the minimum weight of an [ k ]-RDF on  G . In this paper we shall be interested in the study of the [ k ]-Roman domination subdivision number sdof  G  defined as the minimum number of edges that must be subdivided, each once, in order to increase the [ k ]-Roman domination number. We first show that the decision problem associated with sdis NP-hard. Then various properties and bounds are established.

On [ k ]-Roman domination subdivision number of graphs