Dominator semi strong color partition in graphs

Let GG =(V,E)(V,E) be a simple graph. A subset SS is said to be Semi-Strong if for every vertex vv in VV, |N(v)∩S|≤1|N(v)∩S|≤1, or no two vertices of SS have the same neighbour in VV, that is, no two vertices of SS are joined by a path of length two in VV. The minimum cardinality of a semi-strong partition of GG is called the semi-strong chromatic number of GG and is denoted by χsGχsG. A proper colour partition is called a dominator colour partition if every vertex dominates some colour class, that is , every vertex is adjacent with every element of some colour class. In this paper, instead of proper colour partition, semi-strong colour partition is considered and every vertex is adjacent to some class of the semi-strong colour partition.Several interesting results are obtained.