Independent Transversal Domination Number Of A Graph

Let G =(V,E)be a graph.A set S(?)V(G)is a dominating set of G if every vertex in V\S is adjacent to a vertex of S.The domination number of G,denoted by?(G),is the cardinality of a minimum dominating set of G.Furthermore,a dominating set S is an independent transversal dominaling set of G if it intersects every maximum independent set of G.The independent transversal domination number of G.denoted by ?it(G),is the cardinality of a minimum independent transversal dominating set of G.In 2012,Hamid initiated the study of the independent transversal domination of graphs,and posed the following two conjectures:Conjecture 1:If G is a non-complete connected graph on n vertices,then ?it(G)?[n/2].Conjecture 2:If G is a connected bipartite graph,then ?it(G)is either ?(G)or ?(G)+ 1.We show that Conjecture 1 is not true in general.Very recently,Conjecture 2 is partially verified to be true by Ahangar,Samodivkin,Yero.Here,We prove the full statement of Conjecture 2.In addition,we give a correct version of a theorem of Hamid.Finally,we answer a problem posed by Martinez,Almira,and Yero on the independent transversal total domination of a graph.