Total roman domination number of graphs

Abstract

In this paper a characterization of minimal total roman dominating functions has been proved. It has also been proved that if v is a pendant vertex and its neighbor u is not a pendant vertex then f(v) = 0 or 1 for any minimal total roman dominating function f. The operation of vertex removal has been considered. A necessary and sufficient condition under which the total roman domination number increases (under this operation) has been proved. Similarly a necessary and sufficient condition has been proved under which the total roman domination number decreases. Some examples has been given with miscellaneous results. We characterize those graphs for which the total roman domination number is equal to the number of vertices of the graph.