Total Italian Domination: Structural Results on Middle Graphs of Standard Families

If we have graph , an Italian dominating function (IDF) is defined as a mapping satisfying the condition, if for any vertex must be have neighbor there is no fewer than 1 vertex to be or no fewer than 2 vertices where The weight of a (IDF) is the sum of all these functions for all vertices . And the minimum weight selected from overall (IDF) of and indicated by A total Italian dominating function (TIDF) is an (IDF) if the collection of non-zero elements, each of which is contiguous to at least one non-zero element. A minimum weight of any (TIDF) is referred to as a total Italian domination number indicated by . We commenced the study of (TIDF) of graph and the graph can be issued by add a new vertex between every two old vertices in and draw the edges between every two added vertices provided that there exist common vertex binds them, we called middle graph of . For selected standard graph families. The accurate numerical values of these graph-related parameters were established. We determined these parameters' exact values.