On 7-γt-Crictical Graphs

The existence problem of the total domination vertex critical graphs have been widely studied and settled with respect to parities of both the domination number m and the maximum degree △.We will study and give some results for the remaining isolated case for which △=7,9 and 11 for m=7.In this paper,we will present some result for 3-γt-critical graphs with 2≤△≤9.Furthermore,we will provide an alternative proof to show that for m=7 and △=7 there exists no γt-critical graphs.We define H1=x1x2,H2=x3x4 and H3=x5x6 as the components of G[R].Also let Xi=N(Xi)∩ N(v),i=1,2,…6 for which xi∈V(G).We will also present some results graphs with m=7 and △=9.We will show that for |X1|=|X2|= |X3|=|X4|:|X5|=1,|X6|=4 and for |X1|=|X2|=|X3|=|X5|=1,|X4|=2,|X6|=3 there is no 7-γt-critical graphs.Finally we will provide some new results for 7-γt-critical graph with△=11.