A study of line graph theory towards line set domination

Abstract

We introduce the concept of domatic number in LG. The minimum cardinality of vertices in such a set is called a split line domination number in L (G) and is denoted by γsl (G). In this paper, we introduce the new concept in domination theory. Also, we study the graph theoretic properties of γ sl (G) and many bounds were obtained in terms of elements of G and its relationships with other domination parameters were found. For any graph G, the line graph L G H is the intersection graph. Thus the vertices of LG are the edges of G, with two vertices of LG adjacent whenever the corresponding edges of G are. A dominating set D is called independent dominating set of LG, if D is also independent. The independent domination number of LG denoted by i LG, equals min {; DD is an independent dominating set of LG}. Adomatic partition of LG is a partition of VLG, all of whose classes are dominating sets in LG. The maximum number of classes of a domatic partition of LG is called the domatic number of LG and denoted by d LG. In this paper many bounds on i LG were obtained in terms of elements of G, but not in terms of elements of LG. Further we develop its relationship with other different domination parameters.