For a graph G, we call a subset S V [ E a total mixed dominating set of G if each element of V [ E is either adjacent or incident to an element of S, and the total mixed domination number of G is the minimum cardinality of a total mixed dominating set of G. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Study of this number in trees and 2-corona graphs is our next work. Also we show that the total mixed domination number of a graph is equal to the total domination number of its total graph. Giving the total mixed domination numbers of the paths, cycles, complete bipartite graphs, complete graphs and wheels is our last work.
