Strong Split Line Block Domination of a Graph

By a graph we mean a finite, undirected without loops or multiple edges. For any undefined terms or notation, we refer [1]. The line block graph Lb(G) of a graph G is the graph whose vertex set is the union of the set of edges and the set of blocks of G in which two vertices are adjacent if the corresponding blocks are adjacent or one corresponds to a block of G and other to an edge incident with it. This concept was introduced by Kulli [3]. The block cutpoint graph bc (G) of a graph is the graph whose vertex set is the union of set of blocks and the set of cut vertices of G in which two vertices are adjacent if the corresponding blocks are adjacent or the corresponding cutvertices are incident with the blocks. This concept was first studied by Harary in [1] and was studied in [7]. The middle graph M(G) of a graph G is the graph whose vertex set is the union of set of vertices and edges of G with two vertices are adjacent if they are adjacent edges of G on one corresponds to a vertex and the other to an edge incident with it. This concept was introduced in [2] and was studied by Kulli, Patil and Biradar in [4, 5, 6] A set S  V(G) is said to be a dominating set of G, if every vertex in V – S is adjacent to some vertex in S. The minimum cardinality of vertices in such a set is called the domination number of G and is denoted by V (G). A dominating set S  V (G) is a strong split dominating set, if the induced subgraph V – S is totally disconnected with at least two vertices. The strong split domination number ss(G) of G is the minimum cardinality of a strong split dominating set of G. This concept was well studied in [7, 8, 9, 10]. A set D  V [Lb(G)] is said to be strong split line block dominating set if the induced subgraph. V[Lb(G)] – D is totally disconnected with at least two vertices. The strong split line block domination number sslb (G) of Lb (G) is the minimum cardinality of a strong split line block dominating set of G. In this paper, we study the theoretic properties of sslb(G) and many bounds were obtained in terms of elements of G and its relationship with other domination parameters were found. We need the following results for our further results. Theorem A [11] : For any connected (p, q) tree T, ssbc (T) = 2. Theorem B [10] : For any connected (p, q) tree T, sstb (T) = q + 1. Theorem C [9] : For any graph G, ssm (G) = q. Theorem D [7] : For any (p, q) connected graph G, ss (G) = 0 (G).