A dominating set of a graph G which intersects every independent set of maximum cardinality in G is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of G and is denoted by γ it (G). In this paper we study some complexity issues on some independent transversal domination related problems. On the other side, we prove that for every integers a, b, c with a ≤ b ≤ a + c, there exists a graph G such that G has domination number a, minimum degree c and independent transversal domination number b. We also give some other properties of independent transversal dominating sets in graphs.