Total 2 - Out Degree Equitable Domination Number for Distinct Graphs

For any graph G, "The out degree of u with respect to a dominating set D is denoted and defined as <img src=image/13438994_01.gif>". The subset D of V is referred to be a dominating set and the induced sub graph <img src=image/13438994_02.gif> doesn't contain any isolated vertices. The total domination number is determined by the number of vertices of the minimal dominating set, which is denoted by <img src=image/13438994_03.gif>. Based on the above concept of out degree and total dominating set, we introduce a new domination called total 2 - out degree equitable domination (2 - ODED) number. A set D of V is referred to be total 2 – ODED set if D is a dominating set and induced sub graph <img src=image/13438994_02.gif> that contains no isolated vertices also has the property <img src=image/13438994_04.gif> where <img src=image/13438994_01.gif> for any two vertices <img src=image/13438994_05.gif>. The minimum number of vertices of such dominating set is known as total 2 - ODED number. This paper is to investigate the proposed domination number for some general graphs like complete graph, star graph, path graph, cycle graph, double fan graph, bi star, fan graph, helm graph, crown graph, and triangular snake graph which are explained with examples. Finally, we discuss the application of total 2 – out degree equitable domination (2 – ODED) number in real life.