Total Domination And Least Domination In A Tree

Within the last more than twenty years, concurrent with the growth of computer science, graph theory has seen explosive growth. Perhaps the fastest growing area within graph theory is the study of domination in graph. Cockayne and Hedetniemi' s survey on domination appeared in 1997 and contained 20 references. This survey paper seems to have set in motion the modern study of dominating sets in graphs. In 1990, the number of references of this kind has increased quickly from 20 to 400. Seven years later more than 1000 research papers had been published on dominating sets and related sets in graphs , and the field is steadily growing .In this paper , we discuss the relationship between the total domination number and the least domination number . The total domination number of a graph G is the minimum cardinality of a dominating set of Gsuch that every vertex in V has at least one neighbor in D. The least domination number of G is the minimum cardinality of a dominating set of G whose domination number is the minimum. In [4], E.Sampathkumar discussed the relationship among the least domination number , the total domination number and other parameters in a graph. In [2], Odile Favaron pointed out a mistake in [4] and give an upper-bound of .But he also point out this bound perhaps is not the best.In this paper , we first give some sufficient conditions for t in a tree T: Theorem4.1: Let D be a set of T, if the vertices of D are all hooks and eachmiddle vertex of T relative to D has only one neighbor in D, then ji = y, ?Theorem4.2: Let D be a yrset of T, if the vertices of D are all hooks and the connected components of [D]: D1,D2,......, D satisfies | V (Di}|< 3 (l