The Independent Transversal Dombondage Number of a Graph

A set S⊆VS⊆V of vertices in a graph G=(V,E)G=(V,E) is called a dominating set   if every vertex in V−SV−S is adjacent to a vertex in S. I. Sahul Hamid [I. Sahul Hamid, Independent Transversal Domination in Graphs, Discussiones Mathematicae Graph Theory, 32 (2012), 5–17] defined a dominating set which intersects every maximum independent set in G to be 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 γit(G)γit(G). I. Sahul Hamid also defined an independent transversal dombondage number bit(G)bit(G) of a graph G to be the smallest number of edges whose removal from G results in a graph with independent transversal domination number greater than the independent transversal domination number of G or the cardinality of smallest set E of edges for which γit(G−E)>γit(G)γit(G−E)>γit(G). In this paper we initiate a study of this parameter bit(G)bit(G). Sharp bounds for bit(G)bit(G) for trees are obtained and the exact values are determined for several classes of graphs.