TOTAL ROMAN DOMINATION NUMBER OF GRAPHS

TOTAL ROMAN DOMINATION NUMBER OF GRAPHS

D. K. Thakkar, S. M. Badiyani

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Abstract

In this paper a characterization of minimal total roman dominating func- tions 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 dom- inating function f. The operation of vertex removal has been considered. A necessary and su cient condition under which the total roman domination number increases (un- der this operation) has been proved. Similarly a necessary and su cient 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.

Keywords

Total Dominating Set, Total Roman Dominating Function, Total Roman Domination Number, Minimal Total Roman Dominating Function, Minimum Total Roman Dominating Function.